A trail’s rise and horizontal run form a right triangle. To find the sine ratio, first identify the side opposite the marked angle and the hypotenuse, which lies across from the right angle. A familiar side pattern can give you the hypotenuse without Desmos. The tempting slip is to use the horizontal leg in the ratio instead.
Hints
- Hint 1
The hypotenuse is the side across from the right angle. From the marked angle, which leg is across the triangle? That leg is the opposite side.
- Hint 2
The two legs match a scaled -- right triangle. Find the hypotenuse from that pattern, then put the opposite side over it to find the sine ratio.
Step-by-step
Identify the sides and use a familiar triangle
Step 1Name the sides from the angle
No Desmos needed. The legs match a familiar right-triangle side pattern. The right-angle marker at makes the hypotenuse, the side across from the right angle. From at , the -meter rise is opposite the angle. The -meter horizontal leg touches it.
- Step 2
Find the trail’s length
The legs and are three times the legs and of a -- right triangle. Scaling a triangle multiplies every side by the same amount. Scale its hypotenuse by :
So the trail is meters long.
- Step 3
Find the sine ratio
Sine is opposite over hypotenuse, so the -meter rise goes on top, not the -meter horizontal leg:
Divide the top and bottom by :
The sine of the trail’s angle with the horizontal is . Choice A.