for the right triangles below, find the exact values of the side lengths d and b. if necessary, write your…

Answer:

( d = 5\sqrt{2} ) ( b = \frac{8\sqrt{3}}{3} ) (Wait, no, let's re - check the second triangle. Wait, the second triangle has a 30 - 60 - 90 triangle? Wait, the right - angled side is 8? Wait, the original problem: first triangle is a 45 - 45 - 90 triangle, second is 30 - 60 - 90? Wait, no, let's do step by step.

Step1: Find ( d ) in the 45 - 45 - 90 triangle

In a 45 - 45 - 90 right triangle, the legs are equal, and the hypotenuse ( h ) is related to the leg ( l ) by ( h=l\sqrt{2} ). Here, one leg is 5, so the hypotenuse ( d ) (since it's the hypotenuse of the 45 - 45 - 90 triangle) is ( d = 5\sqrt{2} ) (because in a 45 - 45 - 90 triangle, ( \text{hypotenuse}=\text{leg}\times\sqrt{2} ), and the leg is 5).

Step2: Find ( b ) in the 30 - 60 - 90 triangle

Wait, the second triangle: angles are 30°, 60°, 90°. Wait, the side opposite 30° is the shorter leg. Wait, the side with length 8: is it the side adjacent to 30° or opposite? Wait, the right - angled triangle has angles 30°, 60°, 90°. Let's assume the side with length 8 is the side adjacent to 30° (the longer leg). In a 30 - 60 - 90 triangle, the ratio of sides is ( 1:\sqrt{3}:2 ) (shorter leg : longer leg : hypotenuse). Let the shorter leg be ( x ), longer leg be ( x\sqrt{3} ), hypotenuse be ( 2x ). If the longer leg (adjacent to 30°) is 8, then ( x\sqrt{3}=8 ), so ( x = \frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} )? Wait, no, maybe I misread the side. Wait, the problem says "8" on the vertical leg? Wait, maybe the vertical leg is 8, and we need to find ( b ) (the horizontal leg, shorter leg in 30 - 60 - 90? Wait, no, 30° angle: the side opposite 30° is the shortest. Wait, the angle at the bottom is 60°, so the angle at the top is 30°. So the side opposite 30° is ( b ), and the side adjacent to 30° (the vertical leg) is 8. So in a 30 - 60 - 90 triangle, ( \tan(30^{\circ})=\frac{b}{8} ), and ( \tan(30^{\circ})=\frac{1}{\sqrt{3}} ), so ( b = \frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} )? Wait, no, maybe the vertical leg is the side opposite 60°, so ( \sin(60^{\circ})=\frac{8}{\text{hypotenuse}} ), and ( \tan(60^{\circ})=\frac{8}{b} ), ( \tan(60^{\circ})=\sqrt{3} ), so ( b=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} ).

Wait, but let's re - check the first triangle: it's a 45 - 45 - 90 triangle, so legs are equal. One leg is 5, so the other leg is also 5, and hypotenuse ( d=\sqrt{5^{2}+5^{2}}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2} ). That's correct.

For the second triangle: angles are 30°, 60°, 90°. Let's assume the vertical side is 8 (adjacent to 30°), so ( \tan(30^{\circ})=\frac{b}{8} ), ( \tan(30^{\circ})=\frac{1}{\sqrt{3}} ), so ( b = \frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} ). Wait, but maybe the vertical side is the hypotenuse? No, because it's a right triangle, the hypotenuse is the longest side. If the angle is 30°, the hypotenuse is twice the shorter leg. If the vertical side is 8, and it's the hypotenuse, then the shorter leg (opposite 30°) is ( \frac{8}{2}=4 ), and the longer leg (adjacent to 30°) is ( 4\sqrt{3} ). But the problem's second triangle: the angle at the bottom is 60°, so the angle at the top is 30°, so the side opposite 30° is ( b ), and the side adjacent to 30° is 8. So ( \tan(30^{\circ})=\frac{b}{8} ), so ( b = 8\times\frac{1}{\sqrt{3}}=\frac{8\sqrt{3}}{3} ).

So final answers: ( d = 5\sqrt{2} ), ( b=\frac{8\sqrt{3}}{3} )? Wait, no, maybe I misread the length of the vertical side in the second triangle. Wait, the original problem: the second triangle has a vertical side labeled "8"? Wait, the user's image: the second triangle has a vertical leg with length 8? Wait, maybe it's a 30 - 60 - 90 triangle where the side opposite 60° is 8, so:

In a 30 - 60 - 90 triangle, the sides are in the ratio ( 1:\sqrt{3}:2 ) (shorter leg : longer leg : hypotenuse). The longer leg (opposite 60°) is ( x\sqrt{3} ), shorter leg (opposite 30°) is ( x ), hypotenuse is ( 2x ). If the longer leg (opposite 60°) is 8, then ( x\sqrt{3}=8\Rightarrow x=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} ). And ( b ) is the shorter leg? Wait, no, the angle at the bottom is 60°, so the side ( b ) is adjacent to 60°, so it's the shorter leg. Wait, angle at bottom is 60°, so angle at top is 30°, so side ( b ) is opposite 30°, so it's the shorter leg. So ( b=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} ), and the hypotenuse would be ( \frac{16\sqrt{3}}{3} ).

But let's confirm the first triangle: 45 - 45 - 90, legs are 5 and 5, hypotenuse ( d=\sqrt{5^{2}+5^{2}} = 5\sqrt{2} ). Correct.

So:

( d = 5\sqrt{2} )

( b=\frac{8\sqrt{3}}{3} ) (Wait, but maybe the vertical side is 8, and it's the hypotenuse? Then shorter leg is 4, longer leg is ( 4\sqrt{3} ), so ( b = 4\sqrt{3} )? Wait, I think I made a mistake. Let's re - examine the angle labels. The second triangle: right angle at the bottom right, angle at bottom left is 60°, angle at top is 30°. So the sides: the side opposite 30° is ( b ) (bottom side), the side opposite 60° is the vertical side (length 8), and the hypotenuse is the slant side. So in a 30 - 60 - 90 triangle, ( \sin(60^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{8}{h} ), and ( \cos(60^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{h} ). We know that ( \sin(60^{\circ})=\frac{\sqrt{3}}{2} ), so ( \frac{\sqrt{3}}{2}=\frac{8}{h}\Rightarrow h=\frac{16}{\sqrt{3}}=\frac{16\sqrt{3}}{3} ). Then ( \cos(60^{\circ})=\frac{1}{2}=\frac{b}{h}\Rightarrow b=\frac{h}{2}=\frac{8\sqrt{3}}{3} ). So that's correct.

So the values are ( d = 5\sqrt{2} ) and ( b=\frac{8\sqrt{3}}{3} )? Wait, no, maybe the vertical side is 8, and it's the adjacent side to 30°, so ( \tan(30^{\circ})=\frac{b}{8} ), ( \tan(30^{\circ})=\frac{1}{\sqrt{3}} ), so ( b=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} ). Yes, that's consistent.

So:

Answer:

( d = 5\sqrt{2} ) ( b=\frac{8\sqrt{3}}{3} )

Explanation:

Step1: Calculate ( d ) (45 - 45 - 90 triangle)

In a 45 - 45 - 90 right - triangle, the two legs are equal, and the hypotenuse ( c ) is related to a leg ( a ) by the formula ( c = a\sqrt{2} ). For the first triangle, one leg ( a = 5 ). So the hypotenuse ( d=5\sqrt{2} ) (since ( d=\sqrt{5^{2}+5^{2}}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2} )).

Step2: Calculate ( b ) (30 - 60 - 90 triangle)

In a 30 - 60 - 90 right - triangle, the ratio of the sides (shorter leg : longer leg : hypotenuse) is ( 1:\sqrt{3}:2 ). Let the shorter leg (opposite the 30° angle) be ( x ), the longer leg (opposite the 60° angle) be ( x\sqrt{3} ), and the hypotenuse be ( 2x ). In the second triangle, the longer leg (opposite the 60° angle) is 8. So we set up the equation ( x\sqrt{3}=8 ). Solving for ( x ) (which is the length of the shorter leg, and in this case, the shorter leg is ( b ) as it is opposite the 30° angle), we get ( x=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3} ) (rationalizing the denominator: ( \frac{8}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{8\sqrt{3}}{3} )). So ( b = \frac{8\sqrt{3}}{3} ).