Your question: "The shorter diagonal of a rhombus is 50 mm long. Each of its obtuse angles measures 140. Find the length of each side."
To find the length of each side of a rhombus, we can use the properties of a rhombus.
A rhombus is a quadrilateral with four equal sides. It also has two pairs of opposite congruent angles, which means that the sum of each pair of opposite angles is 180 degrees.
In this case, we are given that the shorter diagonal of the rhombus is 50 mm long and each of its obtuse angles measures 140 degrees.
To find the length of each side of the rhombus, we can follow these steps:
Step 1: Draw the rhombus and label the given information.
- Draw a rhombus with sides labeled as "a" and diagonals labeled as "d1" and "d2".
- The shorter diagonal, which is 50 mm long, can be labeled as "d1".
- The obtuse angle, measuring 140 degrees, can be labeled as angle "O".
Step 2: Use the properties of a rhombus to find the lengths of the sides.
- In a rhombus, the diagonals bisect each other at right angles.
- This means that angle "O" is split into two equal angles, each measuring (180 - 140)/2 = 20 degrees.
- Label these two angles as "x", so we have two congruent right triangles with angles 90°, 20°, and 70°.
- We know that the hypotenuse of each right triangle is the length of each side of the rhombus, which is labeled as "a".
Step 3: Use trigonometry to find the length of each side.
- We can use the trigonometric function tangent (tan) to find the lengths of the sides.
- In a right triangle, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
- Since the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side, we have: tan(70°) = a/ (d1/2) => tan(70°) = a/(50/2).
- Solving this equation for "a", we get: a = (50/2) * tan(70°).
- Using a calculator, we can find the value of tan(70°) is approximately 2.747, so a = (50/2) * 2.747.
- Evaluating this expression, we find that a is approximately equal to 68.68 mm.
Therefore, the length of each side of the rhombus is approximately 68.68 mm.