Grade 4 Isosceles and Equilateral Triangles

What this problem type covers

Eighteen problem types across five levels: base and apex angles of an isosceles triangle, the perimeter of a row of joined equilateral triangles, a right triangle carrying an isosceles one, two triangles on the same base, an equilateral and an isosceles triangle on one line, triangles nested by joining midpoints, a triangle whose longest side is a diameter, equilateral triangles built on a joined shape, folded paper where a folded angle equals the angle it covers, counting every triangle in a matchstick lattice, an equilateral triangle after a turn, two equal isosceles triangles overlapped, a square with an isosceles triangle on one side, a run of equal segments along a base line, a square meeting an equilateral triangle, the sizes of triangle a dot array allows, an equilateral triangle around a square, and counting triangles of one size alone.

Example problems

One example from each difficulty band, with worked solutions.

这些题目由系统自动生成。答案在出题时同步确定并经过测试验证，但仍可能存在错误，发现时请告诉我们。

1. 1 · ★1
   Equilateral triangles with sides of 4 cm are joined in a row without overlapping, as shown. What is the perimeter of the shape made from 12 of them?
   
   • ①56 cm
   • ②48 cm
   • ③60 cm
   • ④52 cm
   • ⑤144 cm
   Show solution

   ①56 cm

   1. The edges where two triangles meet are inside the shape, not on its outline.
   2. Joining 12 triangles leaves 14 edges on the outline.
   3. Each edge is 4 cm, so the perimeter is 56 cm.
2. 2 · ★2
   An equilateral triangle and an isosceles triangle stand next to each other on one straight line. The isosceles triangle has an apex angle of 68 degrees. How big is ㉮?
   
   • ①64 degrees
   • ②74 degrees
   • ③56 degrees
   • ④52 degrees
   • ⑤124 degrees
   Show solution

   ①64 degrees

   1. Every angle of an equilateral triangle is 60 degrees.
   2. The isosceles triangle has an apex of 68 degrees, so its base angle is 56.
   3. A straight line is 180 degrees, so ㉮ is 180 − 60 − 56 = 64 degrees.
3. 3 · ★3
   A triangle of paper with two equal sides is folded as shown. The crease meets the base at 64 degrees. How big is ㉮?
   
   • ①128 degrees
   • ②64 degrees
   • ③26 degrees
   • ④52 degrees
   • ⑤116 degrees
   Show solution

   ④52 degrees

   1. A folded angle and the angle it covers are the same size.
   2. Both angles beside the crease are 64 degrees.
   3. A straight line is 180 degrees, so ㉮ is 180 − 64 − 64 = 52 degrees.
4. 4 · ★4
   ㄱㄴㄷㄹ is a square, and ㄱㅁㄹ is an isosceles triangle whose sides ㄹㅁ and ㄹㄱ are equal. Angle ㄱㅁㄹ is 72 degrees. How big is ㉮?
   
   • ①45 degrees
   • ②126 degrees
   • ③63 degrees
   • ④36 degrees
   • ⑤27 degrees
   Show solution

   ⑤27 degrees

   1. ㄱㅁㄹ is isosceles, so angle ㅁㄱㄹ is also 72 and angle ㄱㄹㅁ is 36 degrees.
   2. A corner of the square is 90 degrees, so angle ㅁㄹㄷ is 36 + 90 = 126.
   3. Sides ㄹㅁ and ㄹㄷ are equal, so ㅁㄹㄷ is isosceles as well.
   4. So ㉮ is (180 − 126) ÷ 2 = 27 degrees.
5. 5 · ★5
   10 dots are set out in a triangle. Using the dots as corners, how many different sizes of equilateral triangle can be made?
   
   • ①1
   • ②5
   • ③3
   • ④2
   • ⑤4
   Show solution

   ⑤4

   1. Count the upright triangles by the length of their side first.
   2. There are 3 rows of dots, so 3 upright sizes.
   3. Then count the tilted ones, which are easy to miss.
   4. That makes 4 sizes in all.

Where people usually go wrong

• Halving before subtracting from 180, or forgetting to halve at all.
• Charging three sides for every triangle in a joined row; the joins are inside the shape.
• Answering with the inner base angle when the question asks what is left over.
• Taking the folded angle away once when it sits on both sides of the crease.
• Using 180 where a four-sided shape needs 360.
• Counting only the smallest triangles in a lattice and missing the larger ones.
• Missing the tilted triangles in a dot array, which are the ones that are hard to see.
• Halving the side each step instead of doubling it when working outwards.