The figure shows rhombus 𝐴𝐵𝐶𝐷. Find the measure of angle 𝐵𝐴𝐶 and the length of line segment 𝐴𝐷.
So, here we have a rhombus. Let’s recall the facts we know about a rhombus. A rhombus is a quadrilateral, that’s a four-sided shape, with all four sides the same length, which we could note on the diagram. And since our problem also involves the diagonals of the rhombus, let’s include another key fact about rhombuses. The diagonals of a rhombus are perpendicular bisectors of each other. As they are perpendicular, this means that the four angles formed at the centre will all be 90 degrees. The word bisect means to cut exactly in half.
Looking at the question then, we need to find the measure of angle 𝐵𝐴𝐶. That’s the angle coloured in orange. We now know two angles of this triangle. We have a 30-degree angle and a 90-degree angle. Using the fact that the angles of a triangle add to 180 degrees, we could then find our missing angle, 𝐵𝐴𝐶.
Therefore, the measure of angle 𝐵𝐴𝐶 plus 30 degrees plus 90 degrees will equal 180 degrees. Simplifying this, we have the measure of angle 𝐵𝐴𝐶 plus 120 degrees equals 180 degrees. Subtracting 120 degrees from both sides, we have the measure of angle 𝐵𝐴𝐶 equals 180 degrees subtract 120 degrees. So, the measure of angle 𝐵𝐴𝐶 is 60 degrees.
Next, we need to find the length of line segment 𝐴𝐷. We have already established that the rhombus has four equal sides. And we’re told that the length of 𝐴𝐵 is 29 centimetres. So, our line segment, 𝐴𝐷, must be the same 29 centimetres. So, we have our final answers then, 60 degrees for the measure of angle 𝐵𝐴𝐶 and 29 centimetres for the length of line segment 𝐴𝐷.