Answer:
17. 4/6 = 9/x
21. E(10, 0)
22. E(28/3, 0)
23. E(24, 0)
24. E(9/2, 0)
Explanation:
You want various proportions and measures related to similar triangles.
Similarity
Similar triangles have proportional corresponding side lengths. That is the ratio of sides in one triangle is equal to the ratio of corresponding sides in a similar triangle.
17.
The small triangle and the large one are similar by virtue of their parallel bases. That means the ratios of side lengths are the same. Here, we need to be careful, because the side length of the larger triangle comes in two pieces that need to be added together: 4 +5 = 9.
left side / base = 4/6 = 9/x . . . . . . correct proportion
21–24.
The ratio of x-intercepts is the same as the ratio of y-intercepts. The y-coordinate of point E will always be zero, since point E lies on the x-axis.
E/C = D/B ⇒ E = CD/B
Here we have used B, C, D, E, to represent the non-zero coordinate of each point.
21. E = 8·5/4 = 10. The point is E(10, 0).
22. E = 4·7/3 = 28/3. The point is E(28/3, 0).
23. E = 6·4/1 = 24. The point is E(24, 0).
24. E = 3·9/6 = 9/2. The point is E(9/2, 0).
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Additional comment
Regarding problem 17:
Occasionally, it is useful to use the partial side length in a proportion. For example, if the top and bottom segments on the right side were labeled 7 and y, then we could write the proportion 4/5 = 7/y because top segments 4 and 7 correspond, as do bottom segments 5 and y.
You can do this with the bottom segment as well. Recognizing that 5 is the difference between the top triangle length and the whole triangle length, we can write the proportion for the bottom length by relating its difference from the top triangle length:
4/6 = 5/(x-6) ⇒ 4x-24 = 30 ⇒ x = 54/4 as with the above proportion.