Video Transcript
The diagram shows triangle
𝐴𝐵𝐶. Given that 𝐴𝐷 is perpendicular to
𝐵𝐶, 𝐴𝐷 equals 15 centimeters, 𝐵𝐷 equals 10 centimeters, and 𝐶𝐷 equals seven
centimeters, find the value of the tan of 𝑥 plus 𝑦.
The first thing we want to do is
label our diagram. 𝐴𝐷 equals 15 centimeters, 𝐵𝐷
equals 10 centimeters, and 𝐶𝐷 equals seven centimeters. We’re interested in the tangent of
this angle, the angle 𝑥 plus 𝑦. However, this angle is in a
triangle that’s not a right triangle, so we’ll need a different strategy to solve
for this angle. If we look, we see that angle 𝑥
and angle 𝑦 are both located inside right triangles. This means it’s possible to find
the tangent value of 𝑥 and the tangent value of 𝑦. And by our angle sum identity, we
know that the tan of 𝐴 plus 𝐵 will be equal to the tan of 𝐴 plus the tan of 𝐵
all over one minus the tan of 𝐴 times the tan of 𝐵.
The tangent of any angle in a right
triangle will be equal to the opposite side length over the adjacent side
length. For angle 𝑥, the opposite side
length is 10 and the adjacent side length is 15. The tan of 𝑥 equals 10 over
15. To find the tan of 𝑦, we have the
opposite side length of seven over the adjacent side length of 15. So the tan of 𝑦 equals seven
fifteenths. So the tan of 𝑥 plus 𝑦 is equal
to 10 over 15 plus seven over 15 all over one minus 10 over 15 times seven over
15. Our numerator becomes 17 over
15.
Before we do this multiplying in
the denominator, we can do some simplifying. 10 over 15 simplifies to
two-thirds, and two-thirds times seven fifteenths equals 14 over 45. In our denominator, since we have
one minus 14 over 45, we can rewrite the one as 45 over 45. 45 minus 14 is 31. This means we’re dividing seventeen
fifteenths by 31 over 45. And to divide by a fraction, we
multiply by the reciprocal. We have seventeen fifteenths times
45 over 31. 15 goes into 45 three times. 17 times three equals 51, making
the tan of 𝑥 plus 𝑦 51 over 31.
