Question

Consider the figure below:

a) Use the Pythagorean theorem to find the value of a.
b) Prove that triangles ABE and ACD are similar.
c) Use similar triangles to find the value of x.
d) Find the value of b.
21. Solve for x, y and z

Answer 1

a) a = 13

b) see explanation

c) x = 10

d) b = 13

Explanation:

a) Use the Pythagorean Theorem as directed by the problem.

`\textrm{(leg 1)}^2 + \textrm{(leg 2)}^2 = \textrm{(hypotenuse)}^2`

`12^2 + 5^2 = a^2`

`144 + 25 = a^2`

`√(169) = a`

`13 = a`

a = 13

b) △ABE and △ACD:

- share angle C, and all right angles are congruent.

- share angle A, and any angle is congruent to itself.

- Therefore, they are similar using the AA theorem.

c) To solve for x, compare the ratios of the bottom sides to the right sides.

12 : (12 + 12)

12 : 24 → 24 = 12 · 2

5 : x

x = 5 · 2

x = 10

d) Since we have already found the other two sides of △ACD, we can find its hypotenuse using the Pythagorean theorem and some extra algebra.

LET
`y = a + b`

--------------------------------

`(12 + 12)^2 + x^2 = y^2`

`24^2 + 10^2 = y^2`

`576 + 100 = y^2`

`√(676) = y`

`26 = y`

Now that we have solved for y, we can now plug y into the auxiliary equation from the beginning of step d.

`y = a + b`

`26 = 13 + b`

`13 = b`

b = 13