4. The Tangent Ratio
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Notice that ∠A is common for all the triangles formed and then it does not change its measure.
See solution.
Let's begin by considering â–³ ABC with vertices A(0,0), B(8,5), and C(8,0), which is a right triangle.
Now, let's draw seven perpendicular segments to AC. They will not all intersect at integer values.
Our next step is to complete the following table.
| Ratio | BC/AC | KD/AD | LE/AE | MF/AF | NG/AG | OH/AH | PI/AI | QJ/AJ |
|---|---|---|---|---|---|---|---|---|
| tan A |
Next, we will calculate each of the ratios and round each number to two decimal places.
| Ratio | BC/AC = 5/8 | KD/AD = 4.38/7 | LE/AE = 3.75/6 | MF/AF = 3.28/5.25 | NG/AG = 2.69/4.3 | OH/AH = 1.875/3 | PI/AI = 1.406/2.25 | QJ/AJ = 0.62/1 |
|---|---|---|---|---|---|---|---|---|
| tan A | 0.62 | 0.62 | 0.62 | 0.62 | 0.62 | 0.62 | 0.62 | 0.62 |
As we can see, no matter the length of the segments we drew above, the tangent of ∠A is the same, which makes sense since ∠A is common for all the triangles formed and does not change its measure.