Question

find d.
triangle with 45°, 45°, right angle, hypotenuse 13, side d
write your answer in simplest radical form.
blank units
sqrt button

Explanation:

Step1: Identify triangle type

The triangle is a 45-45-90 right triangle (two 45° angles, one right angle). In a 45-45-90 triangle, the legs are equal, and the hypotenuse \( c \) is related to a leg \( a \) by \( c = a\sqrt{2} \), or \( a=\frac{c}{\sqrt{2}} \). Here, the hypotenuse is \( d \)? Wait, no—wait, the side labeled 13 is a leg? Wait, no, the right angle is between the two legs, and the hypotenuse is opposite the right angle. Wait, looking at the triangle: the two 45° angles are at the left vertex (top and bottom), and the right angle is at the right vertex. So the sides: the left side is \( d \) (the hypotenuse, since it's opposite the right angle? Wait no, no—wait, in a right triangle, the hypotenuse is the side opposite the right angle. So the right angle is at the right, so the hypotenuse is the left side \( d \), and the two legs are the other two sides (one is 13, the other is equal to 13 because it's a 45-45-90 triangle). Wait, no: in a 45-45-90 triangle, the legs are equal, and hypotenuse \( h = leg \times \sqrt{2} \). Wait, let's clarify: the angles are 45°, 45°, 90°, so it's an isosceles right triangle. So the two legs are equal, and the hypotenuse is leg \( \times \sqrt{2} \). Wait, in the diagram, the side labeled 13: is that a leg or the hypotenuse? Let's see: the right angle is at the right, so the two legs are the top and bottom sides (from the right angle to the left vertex's top and bottom), and the hypotenuse is the left side \( d \). Wait, no—wait, the angles at the left vertex are 45° each, so the two legs (the ones forming the right angle) are equal, and the hypotenuse is \( d \). Wait, no, maybe I got it reversed. Let's denote: let the two legs be \( a \) and \( b \), hypotenuse \( c \). In 45-45-90, \( a = b \), \( c = a\sqrt{2} \). Now, in the diagram, the side labeled 13: is that a leg or the hypotenuse? Let's see the angles: the two 45° angles are at the left, so the sides adjacent to the 45° angles: one leg is adjacent to a 45° angle, the other leg is adjacent to the other 45° angle, and the hypotenuse is opposite the right angle. Wait, maybe the side labeled 13 is a leg, and \( d \) is the hypotenuse? Wait, no—wait, if the two angles at the left are 45°, then the triangle is isosceles with the two legs (the ones meeting at the right angle) being equal. So the leg is 13, so hypotenuse \( d = 13\sqrt{2} \)? Wait, no, wait: no, in a 45-45-90 triangle, hypotenuse is leg * sqrt(2). Wait, let's check: if the legs are length \( l \), hypotenuse \( h = l\sqrt{2} \). So if one leg is 13, then hypotenuse \( h = 13\sqrt{2} \). Wait, but in the diagram, the left side is \( d \), which is the hypotenuse? Wait, no, maybe I have the legs and hypotenuse reversed. Wait, let's look again: the right angle is at the right, so the sides: the two legs are the ones from the right angle to the top and bottom of the left vertex, and the hypotenuse is the left side \( d \). So the legs are equal (since angles are 45° each), so each leg is 13? Wait, no—wait, the side labeled 13: is that a leg? So if the leg is 13, then hypotenuse \( d = 13\sqrt{2} \). Wait, that makes sense. Let's confirm: in a 45-45-90 triangle, hypotenuse = leg * sqrt(2). So if leg is 13, hypotenuse is 13√2. So \( d = 13\sqrt{2} \)? Wait, no—wait, maybe I got it wrong. Wait, maybe the side labeled 13 is the hypotenuse, and we need to find the leg? Wait, no, the problem says "Find d", and d is the left side. Wait, let's re-express: in a 45-45-90 triangle, the ratio of sides is \( 1 : 1 : \sqrt{2} \) (leg : leg : hypotenuse). So if the hypotenuse is \( d \), and a leg is 13, then \( d = 13\sqrt{2} \)? Wait, no—wait, no: if the legs are 13 each, then hypotenuse is 13√2. So yes, that's correct. So step 1: identify it's a 45-45-90 triangle (isosceles right triangle). Step 2: apply the hypotenuse formula: hypotenuse = leg × √2. Here, the leg is 13, so hypotenuse d = 13√2.

Step2: Apply 45-45-90 ratio

In a 45-45-90 triangle, hypotenuse \( h = \text{leg} \times \sqrt{2} \). Here, the leg is 13 (since the two legs are equal in a 45-45-90 triangle), so:
\( d = 13 \times \sqrt{2} = 13\sqrt{2} \)

Answer:

\( 13\sqrt{2} \)