Question

what is the perimeter of kite acbo?
31 units
56 units
62 units
64 units

Explanation:

Step1: Identify properties of kite and circle

In kite \(ACBO\), \(OA = OB\) (radii of the circle), so \(OA = OB = 7\). Also, \(AC = BC\) (properties of a kite), and \(AC = 25\) (given), so \(BC = 25\).

Step2: Calculate perimeter of kite

The perimeter of a kite is the sum of all its sides. So perimeter \(P=OA + OB+AC + BC\). Substituting the values: \(P = 7+7 + 25+25\).

Step3: Compute the sum

First, \(7 + 7=14\) and \(25 + 25 = 50\). Then, \(14+50 = 64\)? Wait, no, wait. Wait, maybe I made a mistake. Wait, no, wait, \(OA\) and \(OB\) are radii, so \(OA = OB = 7\), and \(AC = BC = 25\)? Wait, no, wait, in a kite, two pairs of adjacent sides are equal. So \(OA = OB\) (one pair) and \(AC = BC\) (the other pair). So perimeter is \(OA + AC+OB + BC=7 + 25+7 + 25\). Let's calculate that: \(7+25 = 32\), \(7 + 25=32\), then \(32+32 = 64\)? Wait, but the options include 62. Wait, maybe I misread the length. Wait, maybe \(OC\) is 25, and we need to find \(AC\) using Pythagoras? Wait, \(OA\) is 7, \(OC\) is 25, so triangle \(OAC\) is right-angled? Wait, maybe \(OA\) is perpendicular to \(AC\)? Wait, no, in a kite, one diagonal is the perpendicular bisector. Wait, maybe \(OA\) is a radius, so \(OA = 7\), and \(OC\) is 25, so \(AC\) can be found by Pythagoras: \(AC=\sqrt{OC^{2}-OA^{2}}=\sqrt{25^{2}-7^{2}}=\sqrt{625 - 49}=\sqrt{576}=24\). Oh! I see, I made a mistake earlier. So \(AC = BC = 24\), not 25. So let's correct that.

Step1 (corrected): Find length of \(AC\) and \(BC\)

Since \(OA\) is a radius (\(OA = 7\)) and \(OC = 25\), and \(OA\perp AC\) (tangent? Wait, no, \(A\) is on the circle, so \(OA\) is radius, and \(AC\) is a tangent? Wait, if \(A\) is on the circle, then \(OA\) is radius, and \(AC\) is a tangent, so \(OA\perp AC\). So triangle \(OAC\) is right-angled at \(A\). So by Pythagoras, \(AC=\sqrt{OC^{2}-OA^{2}}=\sqrt{25^{2}-7^{2}}=\sqrt{625 - 49}=\sqrt{576}=24\). So \(AC = BC = 24\) (since \(ACBO\) is a kite, \(AC = BC\) and \(OA = OB = 7\)).

Step2 (corrected): Calculate perimeter

Perimeter of kite \(ACBO\) is \(OA + OB+AC + BC\). Substituting values: \(7 + 7+24 + 24\).

Step3 (corrected): Compute the sum

\(7 + 7 = 14\), \(24+24 = 48\), then \(14 + 48=62\).

Answer:

62 units