Step-by-step 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:
3) From geometry, we know that the inner angles of a triangle sum 180°, so we have:
Solving for B, we get:
4) From geometry, we know that Pigatoras Theorem states that:
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:
Solving for x, we get:
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.
Where k is the proportional constant between the sides.
Using the value k = 1.5, we compute the lengths y and z, we get:
Answers
Sides
• x = 12
• y = 10
,
• z = 6
Angles
• A° = 35°
,
• B° = 55°