Two isosceles triangles have equal vertical angles and their areas are in the ratio 4.84 ∶ 5.29. What is the ratio of their corresponding heights?
22 ∶ 23
Finding the Ratio of Corresponding Heights of Similar Isosceles Triangles
This problem asks us to find the ratio of the corresponding heights of two isosceles triangles, given that they have equal vertical angles and the ratio of their areas.
Understanding Isosceles Triangles and Similarity
An isosceles triangle has two sides of equal length and the two angles opposite those sides (base angles) are equal. The third angle is called the vertical angle.
When two isosceles triangles have equal vertical angles, they are similar. Here's why:
- Let the vertical angle of each triangle be \(\alpha\).
- In an isosceles triangle, the sum of angles is \(180^\circ\). If the base angles are \(\beta\), then \(\alpha + 2\beta = 180^\circ\).
- So, \(\beta = \frac{180^\circ - \alpha}{2}\).
- Since both triangles have the same vertical angle \(\alpha\), their base angles \(\beta\) will also be the same.
- Thus, both triangles have the same three angles (AAA similarity criterion), making them similar triangles.
Similar triangles have corresponding angles equal and corresponding sides proportional. Important for this problem, the ratio of their corresponding heights is also equal to the ratio of their corresponding sides.
Relationship Between Area and Height in Similar Triangles
A key property of similar triangles is that the ratio of their areas is equal to the square of the ratio of their corresponding sides, corresponding altitudes (heights), corresponding medians, or corresponding angle bisectors.
Mathematically, if Triangle 1 is similar to Triangle 2, and \(h_1\) and \(h_2\) are their corresponding heights, and Area\(_1\) and Area\(_2\) are their areas, then:
\[ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{h_1}{h_2}\right)^2 \]
Solving the Problem: Calculating the Ratio of Heights
We are given the ratio of the areas of the two isosceles triangles:
\[ \frac{\text{Area}_1}{\text{Area}_2} = \frac{4.84}{5.29} \]
Using the property mentioned above, we can relate this to the ratio of their corresponding heights, \(h_1\) and \(h_2\):
\[ \left(\frac{h_1}{h_2}\right)^2 = \frac{4.84}{5.29} \]
To find the ratio of the heights \(\frac{h_1}{h_2}\), we need to take the square root of the ratio of the areas:
\[ \frac{h_1}{h_2} = \sqrt{\frac{4.84}{5.29}} \]
Let's calculate the square roots:
- \(\sqrt{4.84} = \sqrt{\frac{484}{100}} = \frac{\sqrt{484}}{\sqrt{100}}\)
- We know \(20^2 = 400\) and \(22^2 = 484\). So, \(\sqrt{484} = 22\).
- \(\sqrt{100} = 10\).
- So, \(\sqrt{4.84} = \frac{22}{10} = 2.2\).
- \(\sqrt{5.29} = \sqrt{\frac{529}{100}} = \frac{\sqrt{529}}{\sqrt{100}}\)
- We know \(20^2 = 400\), \(23^2 = 529\). So, \(\sqrt{529} = 23\).
- \(\sqrt{100} = 10\).
- So, \(\sqrt{5.29} = \frac{23}{10} = 2.3\).
Now substitute these values back into the ratio of heights:
\[ \frac{h_1}{h_2} = \frac{2.2}{2.3} \]
To express this as a ratio of whole numbers, we can multiply the numerator and denominator by 10:
\[ \frac{h_1}{h_2} = \frac{2.2 \times 10}{2.3 \times 10} = \frac{22}{23} \]
The ratio of their corresponding heights is 22 : 23.
Checking the Options
Let's compare our calculated ratio with the given options:
- 11 : 23
- 23 : 25
- 22 : 23
- 484 : 529
Our calculated ratio 22 : 23 matches option 3.
| Given Information | Ratio |
|---|---|
| Ratio of Areas | 4.84 : 5.29 |
| Ratio of Corresponding Heights (Calculated) | 22 : 23 |
Conclusion
Since the two isosceles triangles have equal vertical angles, they are similar. The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding heights. By taking the square root of the given area ratio, we found the ratio of the corresponding heights.
Revision Table: Key Concepts for Triangle Ratios
| Property | Relationship in Similar Triangles |
|---|---|
| Ratio of Corresponding Sides (\(s_1 : s_2\)) | \(s_1 : s_2\) |
| Ratio of Corresponding Heights (\(h_1 : h_2\)) | \(h_1 : h_2\) |
| Ratio of Perimeters (\(P_1 : P_2\)) | \(P_1 : P_2 = s_1 : s_2\) |
| Ratio of Areas (Area\(_1\) : Area\(_2\)) | Area\(_1\) : Area\(_2 = (s_1 : s_2)^2 = (h_1 : h_2)^2\) |
Additional Information: Similarity of Triangles Explained
Two triangles are said to be similar if they satisfy any of the following criteria:
- AAA (Angle-Angle-Angle) Similarity: If all three corresponding angles of two triangles are equal. As shown in this problem, if two angles are equal, the third must also be equal, so AA similarity is sufficient.
- SAS (Side-Angle-Side) Similarity: If two sides in one triangle are proportional to two sides in the other triangle, and the included angles are equal.
- SSS (Side-Side-Side) Similarity: If the three sides of one triangle are proportional to the three corresponding sides of the other triangle.
In the case of isosceles triangles with equal vertical angles, the AAA criterion proves their similarity directly because the base angles will also be equal.
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