Part (a)
x = measure of angle BAP
y = measure of angle ABP
Refer to the diagram below.
Notice that for right triangle ABP, we have x+y = 90 which is true for any pair of acute angles of a right triangle.
Since triangle ABC is also a right triangle, and because ABP = y, this must mean angle ACB = x. Use this type of logic to fill in the proper x's and y's shown in the diagram. Each right triangle will get exactly one x and exactly one y for their acute angles.
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Now focus on triangle ABP and triangle CTS.
The previous section gave us:
- angle BAP = angle TCS = x
- angle ABP = angle CTS = y
- angle APB = angle CST = 90
From here we use the angle angle (AA) similarity theorem to conclude that triangle ABP is similar to triangle CTS.
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Part (b)
We're told that segment AS = 4 cm. The tickmark on this segment, and segment SC shows that AS = SC. Then we can say AS+SC = 4+4 = 8 which is the length of side AC.
Triangle ABC is a right triangle with legs AB = 6 and AC = 8. The hypotenuse is CB = 10 after using the pythagorean theorem.
Segment TS splits side AC into two smaller equal pieces (AS and SC). We can think of it as saying SC is half as long as AC.
Due to the intercept theorem, this means the other pieces of triangle CTS are half as long as the corresponding pieces of triangle CBA
- CT = (1/2)*CB = (1/2)*10 = 5
- TS = (1/2)*BA = (1/2)*6 = 3
Since we've proven that triangle ABP was similar to triangle CTS, we can form this proportion below to find BP.
AB/CT = BP/TS
6/5 = BP/3
6*3 = 5*BP
18 = 5*BP
BP = 18/5
BP = 3.6
Then we can say:
BP + PT + TC = BC
3.6 + PT + 5 = 10
PT + 8.6 = 10
PT = 10 - 8.6
PT = 1.4
PT = 7/5