Calculate Trapezoid Perimeter: Inside an Isosceles Triangle with Height 8

Q: Why can I assume that GF = GE and DB = DC?

In an isosceles triangle, the height from the apex to the base creates two congruent right triangles. This means the height bisects the base into two equal parts, so corresponding segments are equal.

Q: How do I know which sides to use in the Pythagorean theorem?

Look for right triangles in the figure! The height AD creates right angles, so use the height as one leg, part of the base as the other leg, and the triangle's side as the hypotenuse.

Q: Why do I need to find FB separately if I already know AB?

Because FB is only part of AB! Since AB = 17 and AF = 5, we get FB = AB - AF = 17 - 5 = 12. Don't use the whole side length when you only need a segment.

Trapezoid Perimeter with Pythagorean Theorem

ABC is an isosceles triangle.

AD is the height of triangle ABC.

AF = 5

AB = 17
AG = 3

AD = 8

What is the perimeter of the trapezoid EFBC?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's determine the perimeter of trapezoid E F B C.

00:09 First, examine all the given information.

00:27 We know that A D is the height of the trapezoid.

00:35 A G is perpendicular to E F.

00:39 Let's apply the Pythagorean theorem to triangle A G F.

00:52 Substitute the values we know to find the length of G F.

01:09 Isolate G F.

01:21 Now, take the square root.

01:39 This gives us the length of G F.

01:46 In an isosceles triangle, the perpendicular line is also a median.

01:55 So, E F is G F plus E G.

02:02 Let's apply the Pythagorean theorem to triangle A D B.

02:16 Substitute the known values to find D B.

02:36 Isolate D B.

02:53 Now, take the square root.

03:07 This is the length of D B.

03:12 Remember, the perpendicular in an isosceles triangle is also a median.

03:17 So, C B is D B plus C D.

03:33 F B equals side A B minus A F.

03:43 Substitute the known values to solve for F B.

03:54 F B is equal to E C because E F intersects the triangle's sides.

04:06 Now, let's calculate the perimeter using all the side lengths.

04:17 The perimeter of the trapezoid is the sum of its sides.

04:21 Substitute the final values to find the perimeter.

04:38 And that’s how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABC is an isosceles triangle.

AD is the height of triangle ABC.

AF = 5

AB = 17
AG = 3

AD = 8

What is the perimeter of the trapezoid EFBC?

Step-by-step solution

To find the perimeter of the trapezoid, all its sides must be added:

We will focus on finding the bases.

To find GF we use the Pythagorean theorem: $A^2+B^2=C^2$ in the triangle AFG

We replace

$3^2+GF^2=5^2$

We isolate GF and solve:

$9+GF^2=25$

$GF^2=25-9=16$

$GF=4$

We perform the same process with the side DB of the triangle ABD:

$8^2+DB^2=17^2$

$64+DB^2=289$

$DB^2=289-64=225$

$DB=15$

We start by finding FB:

$FB=AB-AF=17-5=12$

Now we reveal EF and CB:

$GF=GE=4$

$DB=DC=15$

This is because in an isosceles triangle, the height divides the base into two equal parts so:

$EF=GF\times2=4\times2=8$

$CB=DB\times2=15\times2=30$

All that's left is to calculate:

$30+8+12\times2=30+8+24=62$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Isosceles Property: Height divides base into two equal segments
Pythagorean Theorem: Use $a^2 + b^2 = c^2$ to find missing sides
Verification: Check that EF = 8, CB = 30, and FB = 12 give perimeter 62 ✓

Common Mistakes

Avoid these frequent errors

Forgetting that isosceles triangles have symmetric properties
Don't assume all trapezoid sides are different lengths = missing the equal segments! This leads to wrong calculations for EF and CB. Always remember that in isosceles triangles, the height creates two equal base segments, so GF = GE and DB = DC.

Practice Quiz

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In a right triangle, the side opposite the right angle is called....?

FAQ

Everything you need to know about this question

Why can I assume that GF = GE and DB = DC?

In an isosceles triangle, the height from the apex to the base creates two congruent right triangles. This means the height bisects the base into two equal parts, so corresponding segments are equal.

How do I know which sides to use in the Pythagorean theorem?

Look for right triangles in the figure! The height AD creates right angles, so use the height as one leg, part of the base as the other leg, and the triangle's side as the hypotenuse.

What if I get confused about which segments form the trapezoid?

The trapezoid EFBC has four sides: EF (top base), FB (right side), BC (bottom base), and CE (left side). Draw it carefully and label each side length.

Why do I need to find FB separately if I already know AB?

Because FB is only part of AB! Since AB = 17 and AF = 5, we get FB = AB - AF = 17 - 5 = 12. Don't use the whole side length when you only need a segment.

How can I check if my final answer makes sense?

Add up your four sides: EF + FB + BC + CE = 8 + 12 + 30 + 12 = 62. Also check that your Pythagorean calculations work: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ✓

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Calculate Trapezoid Perimeter: Inside an Isosceles Triangle with Height 8

Trapezoid Perimeter with Pythagorean Theorem

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can I assume that GF = GE and DB = DC?

How do I know which sides to use in the Pythagorean theorem?

What if I get confused about which segments form the trapezoid?

Why do I need to find FB separately if I already know AB?

How can I check if my final answer makes sense?

🌟 Unlock Your Math Potential

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