Calculate Trapezoid Area: Isosceles Triangle with Height 8 and Base 17

Question

Given that the triangle ABC is isosceles,
and inside it we draw EF parallel to CB:

171717888AAABBBCCCDDDEEEFFFGGG53 AF=5 AB=17
AG=3 AD=8
AD the height in the triangle

What is the area of the trapezoid EFBC?

Video Solution

Solution Steps

00:00 Determine the area of trapezoid EFBC
00:03 EF is parallel to CB
00:06 According to corresponding angles, AG is also perpendicular to EF
00:14 Apply the Pythagorean theorem to the triangle AFG
00:18 Apply the theorem in order to determine GF
00:21 Substitute in the relevant values according to the given data
00:28 Isolate GF
00:34 Proceed to solve the powers and then take the square root
00:37 We obtained the length of GF
00:44 Now apply the Pythagorean theorem to the triangle ABD
00:50 Apply the theorem in order to determine DB
00:55 Substitute in the values according to the given data
01:02 Let's isolate DB and take the square root
01:07 This is the size of DB
01:16 In an isosceles triangle, the height is also the median
01:21 Therefore GF is equal to EG
01:27 The same thing happens in the larger triangle ABC
01:30 Therefore DB is equal to CD
01:35 Insert the values into the formula
01:40 Apply the formula for calculating the area of a trapezoid
01:44 (Sum of bases(EF,CB) X height(GD)) ➗ 2
01:50 Base EF = EG + GF
01:54 Base CB = CD + DB
01:58 Height GD = AD - AG
02:06 Substitute in the relevant values into our equation

Step-by-Step Solution

To find the area of the trapezoid, you must remember its formula:(base+base)2+altura \frac{(base+base)}{2}+\text{altura} We will focus on finding the bases.

To find GF we use the Pythagorean theorem: A2+B2=C2 A^2+B^2=C^2  In triangle AFG

We replace:

32+GF2=52 3^2+GF^2=5^2

We isolate GF and solve:

9+GF2=25 9+GF^2=25

GF2=259=16 GF^2=25-9=16

GF=4 GF=4

We will do the same process with side DB in triangle ABD:

82+DB2=172 8^2+DB^2=17^2

64+DB2=289 64+DB^2=289

DB2=28964=225 DB^2=289-64=225

DB=15 DB=15

From here there are two ways to finish the exercise:

  1. Calculate the area of the trapezoid GFBD, prove that it is equal to the trapezoid EGDC and add them up.

  2. Use the data we have revealed so far to find the parts of the trapezoid EFBC and solve.

Let's start by finding the height of GD:

GD=ADAG=83=5 GD=AD-AG=8-3=5

Now we reveal that EF and CB:

GF=GE=4 GF=GE=4

DB=DC=15 DB=DC=15

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

EF=GF×2=4×2=8 EF=GF\times2=4\times2=8

CB=DB×2=15×2=30 CB=DB\times2=15\times2=30

We replace the data in the trapezoid formula:

8+302×5=382×5=19×5=95 \frac{8+30}{2}\times5=\frac{38}{2}\times5=19\times5=95

Answer

95


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