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Find x to Make Triangles Similar Using SSS Similarity Theorem
Mathematics
Grade 10
Question Content
What value of x will make the triangles similar by the SSS similarity theorem? The first triangle has side lengths 15, 15, 21. The second triangle has side lengths 20, 20, x.
Correct Answer
28
Detailed Solution Steps
1
Step 1: Recall the SSS Similarity Theorem: For two triangles to be similar by SSS, the ratios of all corresponding sides must be equal.
2
Step 2: Identify the corresponding sides. The sides of length 15 in the first triangle correspond to the sides of length 20 in the second triangle. The side of length 21 in the first triangle corresponds to the side of length x in the second triangle.
3
Step 3: Calculate the similarity ratio: 20/15 = 4/3. This is the ratio of the sides of the second triangle to the first triangle.
4
Step 4: Set up the proportion for the unknown side: x/21 = 4/3.
5
Step 5: Solve for x by cross-multiplying: x = (21 * 4)/3 = 28.
Knowledge Points Involved
1
SSS Similarity Theorem
The SSS (Side-Side-Side) Similarity Theorem states that if the ratios of the lengths of all three pairs of corresponding sides of two triangles are equal, then the two triangles are similar. This means the triangles have the same shape, with all corresponding angles equal, and their sides are scaled by a constant factor called the similarity ratio.
2
Proportions and Cross-Multiplication
A proportion is an equation stating that two ratios are equal. Cross-multiplication is a method used to solve proportions: if a/b = c/d, then ad = bc. This is used to find unknown values in proportional relationships, such as the side lengths of similar figures.
3
Similarity Ratio
The similarity ratio is the constant factor by which all sides of one similar figure are multiplied to get the side lengths of the other figure. It is calculated by taking the ratio of corresponding sides of the two similar figures, and it applies to all linear measurements of the figures.
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