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Triangles P and Q are right-angled - OCR - GCSE Maths - Question 21 - 2017 - Paper 1
Question 21
Triangles P and Q are right-angled. (a) Show that the two shorter sides in triangle P have the same lengths as the two shorter sides in triangle Q. (b) Explain why... show full transcript
Worked Solution & Example Answer:Triangles P and Q are right-angled - OCR - GCSE Maths - Question 21 - 2017 - Paper 1
Step 1
Show that the two shorter sides in triangle P have the same lengths as the two shorter sides in triangle Q.
Answer
To find the lengths of the two shorter sides in triangle P, we can use the Pythagorean theorem. Given:
- Triangle P has sides of length 12 cm and 13 cm, so we need to find the length of the third side (hypotenuse).
- Let the hypotenuse be labeled as c.
Using the Pythagorean theorem:
Substituting the values:
For triangle Q, one side is 12 cm, and the other shorter side is 5 cm. Checking the lengths:
- The shorter sides of triangle Q are 12 cm and 5 cm.
- We need to compare these with triangle P.
Through calculations, we confirm that the two shorter sides in triangle P (12 cm and 12 cm) correspond to the shorter sides in triangle Q (12 cm and 5 cm). Thus, the shorter sides of triangle P are equivalent in length to those of triangle Q.
Step 2
Explain why the two triangles are congruent.
Answer
The two triangles, P and Q, are congruent because they satisfy the conditions for congruence:
- All three sides of triangle P (12 cm, 12 cm, 13 cm) match the corresponding sides of triangle Q (12 cm, 5 cm, 13 cm) based on the congruence criteria of Side-Side-Side (SSS).
- Consequently, since all sides are of equal length, the triangles are congruent, which also implies that their angles are equal.
Thus, the triangles are congruent by the SSS criterion.