Question

Find the values of x, y, z, A, and B in the image below. Use that facts that both triangles have 90 degree angles, and that the triangles are similar.The value of A and B in degrees? What is the measure of y and z?

Answer 1

Explanation

From the diagram, we see that:

0. the triangles are similar,

,

1. they share a 90° angle.

1) We identify with colour the similar angles and proportional sides:

2) Similar angles have equal lengths, so we have:

$A=35\degree.$

3) From geometry, we know that the inner angles of a triangle sum 180°, so we have:

$\begin{gathered} A+B+90\degree=180\degree, \\ 35\degree+B+90\degree=180\degree. \end{gathered}$

Solving for B, we get:

$B=180\degree-90\degree-35\degree=55\degree.$

4) From geometry, we know that Pigatoras Theorem states that:

$h^2=a^2+b^2.$

Where h is the hypotenuse of a right triangle, a and b are the cathetus.

For the triangle at the left, we see that:

• one cathetus has length a = x (the green side),

,

• the second cathetus has length b = 9 (the blue side),

,

• the hypotenuse has a length h = 15 (the red side).

Replacing these data in the formula above, we have:

$15^2=x^2+9^2.$

Solving for x, we get:

$\begin{gathered} x^2=15^2-9^2=144, \\ x=\sqrt{144}=12. \end{gathered}$

5) We know that the triangles are similar, so their similar sides must have the same proportion, i.e. the quotient of them must be equal to a constant k.

$\begin{gathered} \text{ Red sides: }k=\frac{15}{y}, \\ \text{ Green sides: }k=\frac{x}{8}=\frac{12}{4}=1.5, \\ \text{ Blue sides: }k=\frac{9}{z}. \end{gathered}$

Where k is the proportional constant between the sides.

Using the value k = 1.5, we compute the lengths y and z, we get:

$\begin{gathered} k=\frac{15}{y}=1.5\Rightarrow y=\frac{15}{1.5}=10, \\ k=\frac{9}{z}=1.5\Rightarrow z=\frac{9}{1.5}=6. \end{gathered}$Answers

Sides

• x = 12

• y = 10

,

• z = 6

Angles

• A° = 35°

,

• B° = 55°