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Classify triangles by sides and angles

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5126285
Give the most specific side-based classification for each triangle: equilateral, isosceles, or scalene. Show your reasoning. a) Triangle 1 has side lengths \(5\,\text{cm}\), \(6\,\text{cm}\), and \(7\,\text{cm}\). b) Triangle 2 has side lengths \(a = 6\,\text{cm}\), \(b = 6\,\text{cm}\), and \(c = 6\,\text{cm}\). c) Triangle 3 has side lengths \(a = 4\,\text{cm}\), \(b = 7\,\text{cm}\), and \(c = 4\,\text{cm}\).

Hints

- Compare the three side lengths in each triangle. - How many congruent sides identify an isosceles triangle? - Why is “most specific” important when all three sides are congruent?

Solution

1. For a), all three side lengths are different, so the triangle is scalene. 2. For b), all three sides have length \(6\,\text{cm}\). The triangle is equilateral. 3. For c), two sides have the same length, \(a = c = 4\,\text{cm}\), while the third side has a different length. The triangle is isosceles.

Answer

a) Scalene, because all three side lengths are different. b) Equilateral, because all three sides are congruent. c) Isosceles, because two sides have length \(4\,\text{cm}\).
5372335
The scale drawing shows the large triangle \(EGF\) and segment \(GH\). Segments \(GE\) and \(GF\) are perpendicular. The figure contains three triangles: \(\triangle EGF\), \(\triangle EGH\), and \(\triangle FHG\). One is acute, one is right, and one is obtuse. Match each triangle with its angle classification.
Figure for problem 537233

Hints

- Examine the angles in each of the three triangles. - Use the fact that perpendicular segments form a right angle. - An obtuse angle measures more than \(90^\circ\). - An acute triangle has three acute angles.

Solution

1. In \(\triangle EGF\), the angle at \(G\) is a right angle because \(GE\) and \(GF\) are perpendicular. Therefore, \(\triangle EGF\) is a right triangle. 2. In \(\triangle EGH\), the angle at \(H\) is greater than \(90^\circ\). Therefore, \(\triangle EGH\) is obtuse. 3. All three angles of \(\triangle FHG\) are less than \(90^\circ\). Therefore, \(\triangle FHG\) is acute.

Answer

\(\triangle EGF\): right \(\triangle EGH\): obtuse \(\triangle FHG\): acute
5372245
Rectangle \(KLMN\) is \(8\,\text{cm}\) wide and \(4\,\text{cm}\) high. Diagonals \(\overline{KM}\) and \(\overline{LN}\) intersect at \(S\). a) Name all right triangles formed by two sides of the rectangle and one diagonal. b) Which of the four small triangles—\(\triangle KLS\), \(\triangle LMS\), \(\triangle MNS\), and \(\triangle NKS\)—are obtuse triangles? c) Explain why all four small triangles are isosceles.
Figure for problem 537224

Hints

- Use the right angles at the rectangle’s vertices. - Compare the angles at \(S\) that face the longer and shorter sides. - Recall how the diagonals of a rectangle compare and divide each other. - Identify which sides of each small triangle are half-diagonals.

Solution

1. Each diagonal divides the rectangle into two right triangles. The four right triangles are \(\triangle KLM\), \(\triangle LMN\), \(\triangle MNK\), and \(\triangle NKL\). 2. Because the rectangle is wider than it is high, the angles at \(S\) opposite the longer sides \(\overline{KL}\) and \(\overline{MN}\) are obtuse. Therefore, \(\triangle KLS\) and \(\triangle MNS\) are obtuse triangles. 3. The diagonals of a rectangle are congruent and bisect each other. Thus, \(SK = SL = SM = SN\). Each small triangle has two congruent sides, so all four are isosceles.

Answer

a) \(\triangle KLM\), \(\triangle LMN\), \(\triangle MNK\), and \(\triangle NKL\) b) \(\triangle KLS\) and \(\triangle MNS\) c) The diagonals are congruent and bisect each other, so \(SK = SL = SM = SN\). Each small triangle therefore has two congruent sides.

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