Question

what is the length of z in the diagram?

Explanation:

Step1: Identify similar triangles

The two right - angled triangles are similar. Let the smaller triangle have sides \(a = 5\) and \(b\) (the height), and the larger triangle have sides \(a'=5 + 20=25\) and \(z\) (the hypotenuse of the larger triangle).

Step2: Use the property of similar triangles

If the two right - angled triangles are similar, the ratios of their corresponding sides are equal. Let's assume the ratio of the sides of the smaller triangle to the larger triangle is the same for all corresponding sides. Also, we can use the fact that if we consider the angle \(\theta\) in both triangles, \(\cos\theta\) is the same for similar right - angled triangles. In the smaller triangle, if we assume the adjacent side to the angle \(\theta\) is \(5\) and in the larger triangle, the adjacent side is \(25\).
We know that in a right - angled triangle, if we consider the relationship between the sides and the hypotenuse. Let's assume the triangles are in the ratio \(1:5\) (since \(25\div5 = 5\)).
If we consider the Pythagorean theorem for the smaller triangle, assume the height is \(h\), and for the larger triangle, the height is \(5h\).
Let's use another approach. If we assume the triangles are similar and we know that the ratio of the horizontal sides of the two similar right - angled triangles is \(5:(5 + 20)=1:5\).
The hypotenuse of the smaller triangle and the larger triangle are also in the ratio \(1:5\).
If we assume the hypotenuse of the smaller triangle is \(x\) and the hypotenuse of the larger triangle is \(z\), then \(\frac{x}{z}=\frac{5}{25}\).
Let's assume the triangles are \(30 - 60-90\) triangles (since the ratio of the sides suggests a special right - angled triangle relationship). In a \(30 - 60 - 90\) triangle, if the shorter leg is \(a\), the hypotenuse is \(2a\) for the smaller triangle and for the larger triangle, if the shorter leg is \(5a\), the hypotenuse is \(10a\).
Since the shorter leg of the larger triangle corresponding to the shorter leg of the smaller triangle is \(25\) (where the shorter leg of the smaller triangle is \(5\)), and using the property of similar right - angled triangles, if the shorter leg of the smaller triangle is \(5\) and the shorter leg of the larger triangle is \(25\), and the hypotenuse of the smaller triangle and larger triangle are in the same ratio as the legs.
We know that if we consider the fact that the two triangles are similar, and if we assume the smaller triangle has a horizontal side of \(5\) and the larger triangle has a horizontal side of \(25\), and they are similar right - angled triangles.
The hypotenuse of the larger triangle \(z\) can be found as follows:
Let's assume the triangles are similar and we know that the ratio of the sides of the two triangles is \(1:5\).
If we consider the fact that the triangles are similar right - angled triangles, and we know that the hypotenuse of the larger triangle \(z\) is \(25\).
Let's use the Pythagorean theorem. Assume the height of the smaller triangle is \(h_1\) and for the larger triangle is \(h_2\). Since the triangles are similar, \(h_2 = 5h_1\).
Let the horizontal side of the smaller triangle be \(x_1 = 5\) and of the larger triangle be \(x_2=25\).
By the Pythagorean theorem, for the smaller triangle \(h_1=\sqrt{x_1^{2}-y_1^{2}}\) (where \(y_1\) is the other leg) and for the larger triangle \(h_2=\sqrt{x_2^{2}-y_2^{2}}\).
Since the triangles are similar \(\frac{h_1}{h_2}=\frac{x_1}{x_2}\).
The correct way is to use the property of similar right - angled triangles. The ratio of the sides of the two similar right - angled triangles gives us that if the shorter side of the smaller triangle is \(5\) and the shorter side of the larger triangle is \(25\), and since they are similar, the hypotenuse of the larger triangle \(z = 25\).

Answer:

25