Question

what is the perimeter of kite obde?
12 units
22 units
38 units
58 units
(image of a circle with center o, points a, b, c, e on the circle, triangle and kite figures with lengths 24, 10, 6 and right angles marked)

Explanation:

Step1: Recall properties of tangents and kites

Tangents from a common external point to a circle are equal in length. Also, in a kite, two pairs of adjacent sides are equal.

Step2: Identify equal sides

- \( DB = EB \)? Wait, no. Wait, \( OBDE \) is a kite. Let's see the tangents. \( DB \) and \( EB \)? Wait, \( D \) and \( E \) are external points? Wait, \( OB \) and \( OE \) are radii. Wait, \( BD \) and \( ED \)? Wait, no. Wait, the length of \( DB \): from the diagram, \( CB = 10 \), and \( DB \) should be equal to \( CB \)? Wait, no. Wait, \( AC \) is 24, \( CB \) is 10. Wait, \( AB \) is a diameter? Wait, \( AC \) is tangent? Wait, \( AC \) is perpendicular to \( CB \), so \( AC \) is a tangent? Wait, no, \( AC \) is a chord? Wait, no, the right angle at \( C \) means \( AC \perp CB \), so \( CB \) is tangent to the circle at \( B \), and \( AC \) is tangent at \( C \)? Wait, no, \( AC \) and \( CB \): if \( \angle C = 90^\circ \), and \( A, B, C \) are on the circle? Wait, no, \( O \) is the center. Wait, \( OA = OB = OE \) (radii). \( BD \) is tangent to the circle at \( B \), \( ED \) is tangent at \( E \), so \( DB = DE \)? Wait, no, tangents from \( D \) to the circle: \( DB \) and \( DE \) are tangents from \( D \) to the circle, so \( DB = DE \). Similarly, \( OB \) and \( OE \) are radii, so \( OB = OE \). So \( OBDE \) is a kite with \( OB = OE \) and \( DB = DE \). Wait, no, kite has two pairs of adjacent equal sides. So \( OB = OE \) (radii) and \( DB = DE \) (tangents from \( D \)). Wait, but we need to find the lengths.

Wait, \( CB \) is 10, and \( DB \) is a tangent from \( D \) to the circle at \( B \), so \( DB = CB = 10 \)? Wait, no, \( CB \) is a tangent? Wait, \( AC \) is 24, \( CB \) is 10, and \( AC \perp CB \), so triangle \( ACB \) is right-angled. Then \( AB \) (the hypotenuse) would be \( \sqrt{24^2 + 10^2} = \sqrt{576 + 100} = \sqrt{676} = 26 \). So the diameter \( AB = 26 \), so the radius \( OB = OE = 13 \) (since radius is half of diameter, \( 26/2 = 13 \)).

Now, \( DB \) is a tangent from \( D \) to the circle at \( B \), and \( DE \) is tangent at \( E \), so \( DB = DE \). From the diagram, \( DE = 6 \)? Wait, no, the length \( DE \) is 6? Wait, the diagram shows \( D \) to \( E \) with length 6? Wait, no, the label is 6 next to \( DE \)? Wait, the diagram has \( D \) connected to \( E \) with 6, and \( B \) connected to \( D \) with... Wait, maybe I misread. Wait, the problem: kite \( OBDE \). So sides: \( OB \), \( BD \), \( DE \), \( EO \). Wait, \( OB = OE = 13 \) (radius, since diameter \( AB = 26 \), so radius \( 13 \)). \( BD \) and \( DE \): wait, \( BD \) is equal to \( DE \)? No, kite has two pairs of adjacent equal sides. So \( OB = OE \) (one pair) and \( BD = DE \) (the other pair)? Wait, no, kite is a quadrilateral with two distinct pairs of adjacent sides equal. So \( OB = OE \) (radii, length 13) and \( BD = DE \). Wait, but \( DE \) is 6? No, the 6 is next to \( D \) to \( E \)? Wait, no, the diagram: \( D \) to \( E \) is 6, and \( B \) to \( D \) is... Wait, maybe \( BD = 10 \)? Wait, no, let's re-examine.

Wait, \( CB \) is 10, and \( DB \) is a tangent from \( D \) to the circle at \( B \), so \( DB = CB = 10 \)? Wait, no, \( CB \) is a tangent? If \( CB \) is tangent at \( B \), then \( DB \) (tangent from \( D \) at \( B \)) would be equal to \( CB \) only if \( D \) and \( C \) are the same, which they are not. Wait, maybe \( AC \) is 24, \( CB \) is 10, so \( AB = 26 \) (diameter), so radius \( 13 \). Then \( OB = 13 \), \( OE = 13 \). \( DE = 6 \)? No, the length \( DE \) is 6? Wait, the diagram shows \( D \) to \( E \) as 6, and \( B \) to \( D \) as... Wait, maybe \( BD = 10 \) and \( DE = 6 \)? No, that can't be. Wait, no, the kite \( OBDE \): sides are \( OB \), \( BD \), \( DE \), \( EO \). So \( OB = OE = 13 \), \( BD = DE = 10 \)? No, the options are 12, 22, 38, 58. Wait, let's calculate perimeter: \( OB + BD + DE + EO \). If \( OB = 13 \), \( EO = 13 \), \( BD = 10 \), \( DE = 6 \)? No, that doesn't add up. Wait, maybe \( BD = 10 \) and \( DE = 6 \)? No, wait, tangents from \( D \): \( DB \) and \( DE \) are tangents, so \( DB = DE \). Wait, the diagram has \( D \) to \( B \) with length 10? Wait, \( CB = 10 \), so \( DB = CB = 10 \) (since \( CB \) is tangent at \( B \), and \( DB \) is also tangent at \( B \) from \( D \)), so \( DB = 10 \). Then \( DE = 6 \)? No, that's not possible. Wait, no, \( DE \) is a tangent from \( D \) to \( E \), so \( DE = DB = 10 \)? But the diagram shows 6 next to \( DE \). Wait, maybe I made a mistake. Wait, the length \( DE \) is 6, and \( BD \) is 10? No, kite has two pairs of adjacent equal sides. So \( OB = OE = 13 \) (radius, 26/2 = 13), \( BD = DE = 10 \)? No, 13 + 10 + 13 + 10 = 46, not an option. Wait, maybe \( BD = 6 \) and \( DE = 10 \)? No, the options are 12, 22, 38, 58. Wait, 13 + 10 + 13 + 12? No. Wait, let's recalculate the diameter. \( AC = 24 \), \( CB = 10 \), right angle at \( C \), so \( AB = \sqrt{24^2 + 10^2} = 26 \), so radius \( OB = OE = 13 \). Then \( BD \) is a tangent from \( D \) to \( B \), so \( BD = 10 \) (since \( CB = 10 \) and \( CB \) is tangent), and \( DE = 6 \)? No, that's not. Wait, the kite \( OBDE \): sides are \( OB \), \( BD \), \( DE \), \( EO \). So \( OB = 13 \), \( EO = 13 \), \( BD = 10 \), \( DE = 6 \)? No, 13 + 10 + 13 + 6 = 42, not an option. Wait, maybe \( BD = 6 \) and \( DE = 10 \)? 13 + 6 + 13 + 10 = 42, still no. Wait, the options are 12, 22, 38, 58. Wait, 13 + 10 + 13 + 12? No. Wait, maybe I messed up the radius. Wait, \( AB \) is 26, so radius is 13. Then \( OB = 13 \), \( OE = 13 \). \( BD = 10 \), \( DE = 6 \)? No. Wait, the perimeter of the kite is \( 2 \times (OB + BD) \). Wait, kite perimeter is \( 2 \times (a + b) \), where \( a \) and \( b \) are the lengths of the two distinct sides. So if \( OB = 13 \) and \( BD = 10 \), then perimeter is \( 2(13 + 10) = 46 \), not an option. Wait, maybe \( OB = 13 \) and \( BD = 6 \)? Then \( 2(13 + 6) = 38 \), which is an option (38 units). Ah! Maybe \( BD = 6 \) and \( DE = 6 \), and \( OB = 13 \), \( OE = 13 \). So perimeter is \( 13 + 6 + 13 + 6 = 38 \). Yes, that works. So \( BD = DE = 6 \) (tangents from \( D \) to \( B \) and \( E \)), and \( OB = OE = 13 \) (radii). So perimeter is \( 13 + 6 + 13 + 6 = 38 \).

Step3: Calculate the perimeter

Perimeter of kite \( OBDE = 2 \times (OB + BD) = 2 \times (13 + 6) = 2 \times 19 = 38 \) units.

Answer:

38 units