3) The sides of a triangle are equal to the sides of another triangle.
i) Is the length of the height from each side of one triangle to the opposite vertex equal to the height from the equal side of the other triangle to its opposite vertex? Why?
ii) Are the areas of the two triangles equal? Why?
Answer (2)
The heights from corresponding sides of two triangles with equal sides are equal due to the equal angles from the equal sides. Similarly, the areas of the two triangles are also equal as they can be calculated with the same base and height. Therefore, both the heights and areas of the triangles are equivalent.
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When two triangles have sides that are equal in length, they are considered congruent by the Side-Side-Side (SSS) Congruence Theorem. Let's address each part of the question:
i) Is the length of the height from each side of one triangle to the opposite vertex equal to the height from the equal side of the other triangle to its opposite vertex? Why?
In congruent triangles, corresponding parts are equal. Therefore, the heights drawn from corresponding sides to opposite vertices will be equal in length. This is because congruence implies that all corresponding angles are also equal, which in turn means that corresponding heights (perpendicular distances from the base to the opposite vertex) are equal.
ii) Are the areas of the two triangles equal? Why?
Yes, the areas of the two triangles are equal. When triangles are congruent, not only are their corresponding sides and angles equal, but their corresponding geometric properties, such as area, are also equal. The area of a triangle can be calculated using the formula:
Area = 2 1 × base × height
Since the base and the height of corresponding sides in congruent triangles are equal, the calculated areas will be identical for both triangles.
To summarize, if two triangles have sides of equal lengths, they are congruent. This congruence implies that both the heights and the areas are equal because all corresponding parts of congruent triangles are the same.
