Area of a Triangle: Calculate The Missing Side based on the formula

The formula to calculate the area of a triangle is:

(side * height descending from the side) /2

We place the data we have into the formula to find X:

$20=\frac{AB\times AC}{2}$

$20=\frac{x\times5}{2}$

Multiply by 2 to get rid of the fraction:

$5x=40$

Divide both sections by 5:

$\frac{5x}{5}=\frac{40}{5}$

$x=8$

The problem provides the area of a right triangle $ABC$, which is 40, and tells us that $AB = 10$, one of the legs. We need to find the base $BC$ of the triangle.

To find $BC$, we use the formula for the area of a triangle:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Here, the area is 40, the height $AB$ is 10, and the base is $BC$:

$40 = \frac{1}{2} \times BC \times 10$

We can simplify this equation to solve for $BC$:

$40 = 5 \times BC$
$BC = \frac{40}{5}$
$BC = 8$

Hence, the length of side $BC$ is $\boxed{8}$.

To determine the height $h$ of triangle DEF given its area is 70 cm² and the side FE is 14 cm, we follow these steps:

• Identify the known elements: The base $b$ is 14 cm and the area $S$ is 70 cm².
• Use the area formula for a triangle:
  $S = \frac{1}{2} \times b \times h$.
• Rearrange the formula to solve for $h$:
  $h = \frac{2 \times S}{b}$.
• Substitute the given values into the formula:
  $h = \frac{2 \times 70}{14}$.
• Simplify the calculation:
  $h = \frac{140}{14} = 10$ cm.

Therefore, the height $h$ of triangle DEF is $10$ cm.

To find the length of side $BC$, follow these steps:

Step 1: Identify the given information

• The area of the triangle is given as $10$.
• The height $AB$ is $4$.

Step 2: Apply the area formula for a right triangle

The formula for the area $A$ of a triangle is:

$A = \frac{1}{2} \times \text{base} \times \text{height}$

Step 3: Set up the equation

Substituting the known values into the formula gives:

$10 = \frac{1}{2} \times BC \times 4$

Step 4: Solve for $BC$

Begin by simplifying the equation:

$10 = 2 \times BC$

Dividing both sides by 2 to solve for $BC$, we obtain:

$BC = \frac{10}{2} = 5$

Therefore, the length of side $BC$ is $5$.

To solve for the length of side $BC$ in the right triangle $\triangle ABC$, we start with the formula for the area of a right triangle:

$\text{Area} = \frac{1}{2} \times AB \times BC$

We know the area of the triangle is 36, and the length of side $AB$ is 12. We'll substitute these values into the formula:

$36 = \frac{1}{2} \times 12 \times BC$

To isolate $BC$, first multiply both sides of the equation by 2 to eliminate the fraction:

$72 = 12 \times BC$

Now, solve for $BC$ by dividing both sides of the equation by 12:

$BC = \frac{72}{12}$

Upon simplifying, we find:

$BC = 6$

Thus, the length of side $BC$ is $\boxed{6}$.

To solve for the length of side $BC$ in the right triangle, we will use the area formula for triangles:

• Step 1: Identify the given elements: $\text{Area} = 10.5$ and one leg $AB = 3$.
• Step 2: Use the formula for the area of a right triangle:
  $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
• Step 3: Substitute the known values into the formula:
  $10.5 = \frac{1}{2} \times 3 \times BC$
• Step 4: Solve for $BC$:
  Multiply both sides by 2 to clear the fraction:
  $21 = 3 \times BC$
• Step 5: Divide both sides by 3 to isolate $BC$:
  $BC = \frac{21}{3} = 7$

Therefore, the length of side $BC$ is $\boxed{7}$.

To solve the problem, we start by identifying that the area of a right triangle is given by the formula:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Given: $\text{Area} = 21$ and one leg of the triangle, say the height $AB = 7$.

We denote the other leg, which we need to find, as $BC$. Thus:

$21 = \frac{1}{2} \times 7 \times BC$

Solving for $BC$, first multiply both sides by 2 to isolate the product of $7$ and $BC$:

$42 = 7 \times BC$

Now, divide both sides by 7 to solve for $BC$:

$BC = \frac{42}{7} = 6$

Therefore, the length of side $BC$ is $\mathbf{6}$.

To solve this problem, let's apply the formula for the area of a triangle:

The area $A$ of a right triangle can be expressed as

Let's denote side BC (the base) as $b$ and the given height AB as 2. Substituting in the known values, we have:

Simplifying the right side gives us:

Therefore, the length of side BC is 7 units.

To solve this problem, we'll follow these steps:

• Step 1: Use the given information to set up the equation for the area of the triangle.

• Step 2: Calculate the length of side $BC$ using the area formula.

• Step 3: Verify the solution with the given choices.

Now, let's work through each step:
Step 1: We know the area of the right triangle $\triangle ABC$ is given as $27$. The formula for the area of a right triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
Given that $AB = 9$ can be considered as the base, let $BC$ be the height. Thus, the area formula translates to: $27 = \frac{1}{2} \times 9 \times BC$

Step 2: We solve for $BC$ by rearranging the formula:
$27 = \frac{1}{2} \times 9 \times BC$
$27 = 4.5 \times BC$
$BC = \frac{27}{4.5}$
$BC = 6$

Step 3: According to the calculation, the length of $BC$ is $6$. Reviewing the choices given, the correct answer is option 1: $6$.

Therefore, the length of side $BC$ is $6$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the area formula related to a right triangle.
• Step 2: Set up an equation with the given area and known side.
• Step 3: Solve for the unknown side $BC$.

Step 1: We know the area $A$ of a right triangle is given by the formula:
$A = \frac{1}{2} \times \text{base} \times \text{height}$.

Step 2: Using the known values, $A = 38 \, \text{cm}^2$, the side $\text{AC} = 8 \, \text{cm}$ and assuming it acts as the base, we set up the equation:
$38 = \frac{1}{2} \times 8 \times \text{BC}$.

Step 3: Simplify to solve for $\text{BC}$:
Multiply both sides by 2 to eliminate the fraction:
$76 = 8 \times \text{BC}$.
Now, divide both sides by 8 to find $\text{BC}$:
$\text{BC} = \frac{76}{8}$.
$\text{BC} = 9.5 \, \text{cm}$.

Therefore, the length of side $BC$ is 9.5 cm.

To solve this problem, we'll use the formula for the area of a triangle:

• Step 1: Write the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
• Step 2: Substitute the given values: $60 = \frac{1}{2} \times 12 \times \text{DH}$.
• Step 3: Simplify the equation by calculating the half of the base: $\frac{1}{2} \times 12 = 6$.
• Step 4: Replace and solve the equation: $60 = 6 \times \text{DH}$.
• Step 5: Isolate $\text{DH}$ by dividing both sides by 6: $\text{DH} = \frac{60}{6}$.
• Step 6: Calculate the result: $\text{DH} = 10$.

The height from point D to the base FE, $\text{DH}$, is 10 cm.

To solve this problem, we need to calculate the length of side $BC$ in triangle $\triangle ABC$ given that the area is 32 and side $AB = 8$.

We start by using the area formula for a right triangle:

In this context, the base $AB$ is 8, and the height $BC$ is the unknown we need to find. Thus, we have:

We can simplify this equation:

Now, solve for $BC$ by dividing both sides of the equation by 4:

Therefore, the length of side $BC$ is $\mathbf{8}$.

Thus, the solution to the problem is $BC = 8$.

We can insert the given data into the formula in order to calculate the area of the triangle:

$S=\frac{AD\times BC}{2}$

$20=\frac{8\times BC}{2}$

Cross multiplication:

$40=8BC$

Divide both sides by 8:

$\frac{40}{8}=\frac{8BC}{8}$

$BC=5$

We use the formula to calculate the area of the triangle.

Pay attention: in an obtuse triangle, the height is located outside of the triangle!

$$$\frac{Side\cdot Height}{2}=\text{Triangular Area}$

Double the equation by a common denominator:

$\frac{4\cdot PQ}{2}=30$

$\cdot2$

Divide the equation by the coefficient of $PQ$.

$4PQ=60$ / $:4$

$PQ=15$

To find $x$, the vertical height of the triangle, we will use the area formula for a triangle:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

We know that:

• The area of the triangle is 15.
• The base of the triangle, $BE$, is 5 units.
• The height of the triangle, $AE$, is $x$ units.

Substituting these values into the formula, we get:

$15 = \frac{1}{2} \times 5 \times x$

First, simplify the right side of the equation:

$15 = \frac{5}{2} \times x$

To isolate $x$, multiply both sides by 2:

$30 = 5x$

Finally, divide both sides by 5 to solve for $x$:

$x = \frac{30}{5} = 6$

Therefore, the value of $x$ is 6.

To find the missing side $X$ of the triangle:

• Step 1: Identify the given values: area $A = 18.5$ and height $h = 10$.
• Step 2: Use the area formula for a triangle: $A = \frac{1}{2} \times \text{base} \times \text{height}$.
• Step 3: Plug in the known values into the formula:
  $18.5 = \frac{1}{2} \times X \times 10$.
• Step 4: Simplify and solve for $X$:
  $18.5 = 5 \times X$
  Divide both sides by 5 to isolate $X$:
  $X = \frac{18.5}{5}$.
• Step 5: Calculate $X$:
  $X = 3.7$.

Therefore, the missing side $X$ is $3.7$.

To solve for $x$, let's apply the standard formula for the area of a triangle:

• Given that the area $A = 15$, base $b = 3$, and height $h = x$.

The area formula is:

$A = \frac{1}{2} \times b \times h$

Substituting the given values into the equation, we have:

$15 = \frac{1}{2} \times 3 \times x$

Now, simplify and solve for $x$:

$15 = \frac{3}{2} \times x$

Multiply both sides by $\frac{2}{3}$ to isolate $x$:

$x = 15 \times \frac{2}{3}$

Calculating, we obtain:

$x = \frac{30}{3} = 10$

Thus, the height $x$ of the triangle is $x = 10$.

Therefore, the solution to the problem is $x = 10$.

To solve for $x$, we begin by applying the formula for the area of a triangle:

Given: the area is 18, AE is the height (6) , and EC is the x.

Insert the known values into the formula:

Simplify the equation:

Next, solve for $x$ by dividing both sides by 3:

Calculate:

Thus, the length $x$ is $\mathbf{6}$.

To solve this problem, let's apply the following steps:

• Step 1: Identify the formula for the area of a triangle, which is $A = \frac{1}{2} \times \text{base} \times \text{height}$.
• Step 2: Substitute the known values into the formula: $21 = \frac{1}{2} \times 7 \times x$.
• Step 3: Simplify and solve the equation for $x$.

Now, let's work through each step more precisely:
Step 1: We're given the area formula as $A = \frac{1}{2} \times b \times h$.
Step 2: Substitute in the known values: the area $A = 21$, the base $b = 7$, and the height $h = x$, leading to the equation $21 = \frac{1}{2} \times 7 \times x$.
Step 3: Solve for $x$ – first simplify the multiplication on the right: $21 = \frac{7}{2} \times x$.
Step 4: To isolate $x$, multiply both sides by 2 to get $42 = 7x$.
Step 5: Finally, divide both sides by 7 to solve for $x$: $x = \frac{42}{7} = 6$.

Therefore, the value of $x$ is $6$.

To solve this problem, let's follow these steps:

• Step 1: Identify the given information: The area $A = 8.375$, one side $a = X$, and the other side $b = 2.5$.
• Step 2: Utilize the formula for the area of a right triangle, $A = \frac{1}{2} \times \text{base} \times \text{height}$.
• Step 3: Plug in the known values to the formula and solve for $X$.

Detailed solution:

We have the area formula for a right triangle:

Substitute the given area value:

Let's rearrange this equation to solve for $X$:

Calculate:

Therefore, the length of side $X$ is $\mathbf{6.7}$.

This corresponds to choice 2: 6.7