Answer:
The answer is b: 25
Note however that the shape of the diagram does not match the numbers on it. So either it is poorly drawn, or I am misinterpreting the numbers. It seems pretty clear though that the image is incorrect, as the two segments both marked as being 25 units long are represented at different lengths.
Step-by-step explanation:
The image here does not fit the numbers. Note that the bottom triangle has two sides with given lengths of 25 units, but they are drawn as different lengths.
Let's plough on though, and find a better image of what that shape would be.
Starting with the bottom triangle:
Given that the two sides of the bottom triangle are both 25 units in length, we know that it is an isosceles triangle (despite its depiction otherwise). Because of that, we know that the two bottom angles are equal. We already know that the top corner is 90°, and that the sum of the angles in a triangle must be 180° This means that the two bottom angles (again, contrary to how its drawn) are both equal to 45°
Now that we have that information, we can look at the over triangle. We know that the bottom right corner is 45°, and that the bottom left is 90°. That tells us that the top angle is also 45°.
With that said, we can redraw the image with the proper proportions. See the attached image.
That very much simplifies things. The line separating the triangles bisects the hypotenuse of the larger triangle, meaning that x is also equal to 25.
To be sure of that though, we can prove it another way. Given that the bottom two corners are both 45, we can use the sine rule to measure x. The sine rule states that the sine of each angle in a triangle has the same ratio to their opposite sides. Usually expressed as:
[tex]\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}[/tex]
where a, b and c are the side lengths, and A, B and C are the angles of the corners opposite those sides.
With that we can take the two parts of the overall triangle's hypotenuse, 25 and x, and say that each of those divided by the sine of the angle opposite them is equal to each other:
[tex]\frac{x}{\sin 45} = \frac{25}{\sin 45}[/tex]
This turns out to be very easy to solve, as both sides have the same denominator. As a result, we can just multiply both sides by sin 45, giving us:
x = 25.
