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special right triangles isosceles right triangle 30-60-90 triangle find…

Question

special right triangles
isosceles right triangle
30-60-90 triangle
find the missing sides.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
cross out the correct answers. the remaining letters (one per space) complete the statement.
in a 30-60-90 degrees right triangle, the side opposite the 30-degree angle is

Explanation:

Solve problems 1 to 3

Using the 45-45-90 Triangle Theorem and 30-60-90 Triangle Theorem knowledge points

  • Problem 1: This is a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with a leg of length \(4\).
  • Leg \(a = 4\)
  • Hypotenuse \(h = a\sqrt{2} = 4\sqrt{2}\)
  • Missing sides: \(4\) and \(4\sqrt{2}\)
  • Problem 2: This is a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse \(10\).
  • Hypotenuse \(2a = 10 \implies a = 5\) (opposite \(30^\circ\))
  • Long leg \(a\sqrt{3} = 5\sqrt{3}\) (opposite \(60^\circ\))
  • Missing sides: \(5\) and \(5\sqrt{3}\)
  • Problem 3: This is an isosceles right triangle (\(45^\circ\)-\(45^\circ\)-\(90^\circ\)) with hypotenuse \(5\sqrt{2}\).
  • Hypotenuse \(a\sqrt{2} = 5\sqrt{2} \implies a = 5\)
  • Missing sides: \(5\) and \(5\)

Solve problems 4 to 6

Using the 45-45-90 Triangle Theorem and 30-60-90 Triangle Theorem knowledge points

  • Problem 4: This is a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with short leg \(4\) (opposite \(30^\circ\)).
  • Short leg \(a = 4\)
  • Long leg \(a\sqrt{3} = 4\sqrt{3}\)
  • Hypotenuse \(2a = 8\)
  • Missing sides: \(4\sqrt{3}\) and \(8\)
  • Problem 5: This is a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with hypotenuse \(3\sqrt{2}\).
  • Hypotenuse \(a\sqrt{2} = 3\sqrt{2} \implies a = 3\)
  • Missing sides: \(3\) and \(3\)
  • Problem 6: This is a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with long leg \(4\sqrt{3}\) (opposite \(60^\circ\)).
  • Long leg \(a\sqrt{3} = 4\sqrt{3} \implies a = 4\)
  • Hypotenuse \(2a = 8\)
  • Missing sides: \(4\) and \(8\)

Solve problems 7 to 9

Using the 45-45-90 Triangle Theorem and 30-60-90 Triangle Theorem knowledge points

  • Problem 7: This is a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse \(6\).
  • Hypotenuse \(2a = 6 \implies a = 3\) (opposite \(30^\circ\))
  • Long leg \(a\sqrt{3} = 3\sqrt{3}\) (opposite \(60^\circ\))
  • Missing sides: \(3\) and \(3\sqrt{3}\)
  • Problem 8: This is a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with long leg \(9\) (opposite \(60^\circ\)).
  • Long leg \(a\sqrt{3} = 9 \implies a = \frac{9}{\sqrt{3}} = 3\sqrt{3}\)
  • Hypotenuse \(2a = 6\sqrt{3}\)
  • Missing sides: \(3\sqrt{3}\) and \(6\sqrt{3}\)
  • Problem 9: This is a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with hypotenuse \(6\).
  • Hypotenuse \(a\sqrt{2} = 6 \implies a = \frac{6}{\sqrt{2}} = 3\sqrt{2}\)
  • Missing sides: \(3\sqrt{2}\) and \(3\sqrt{2}\)

Solve problems 10 to 12

Using the 45-45-90 Triangle Theorem and 30-60-90 Triangle Theorem knowledge points

  • Problem 10: This is a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with leg \(2\sqrt{2}\).
  • Leg \(a = 2\sqrt{2}\)
  • Hypotenuse \(a\sqrt{2} = 2\sqrt{2} \cdot \sqrt{2} = 4\)
  • Missing sides: \(2\sqrt{2}\) and \(4\)
  • Problem 11: This is a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with leg \(5\).
  • Leg \(a = 5\)
  • Hypotenuse \(a\sqrt{2} = 5\sqrt{2}\)
  • Missing sides: \(5\) and \(5\sqrt{2}\)
  • Problem 12: This is a \(30^\…

Answer:

No.Missing Sides
2\(5\), \(5\sqrt{3}\)
3\(5\), \(5\)
4\(4\sqrt{3}\), \(8\)
5\(3\), \(3\)
6\(4\), \(8\)
7\(3\), \(3\sqrt{3}\)
8\(3\sqrt{3}\), \(6\sqrt{3}\)
9\(3\sqrt{2}\), \(3\sqrt{2}\)
10\(2\sqrt{2}\), \(4\)
11\(5\), \(5\sqrt{2}\)
12\(6\), \(6\sqrt{3}\)

Completed Statement:
In a 30-60-90 degrees right triangle, the side opposite the 30-degree angle is half the hypotenuse.