Question

read the problem below. decide whether gary or grace is correct
find the ( mangle klm ).
gary thinks to find the ( mangle klm ) he should use this formula:
( mangle klm = \frac{1}{2} left( moverarc{jn} - moverarc{km}
ight) )
grace thinks to find the ( mangle klm ) she should use this formula:
( mangle klm = \frac{1}{2} left( moverarc{jn} + moverarc{km}
ight) )
(circ) grace
(circ) gary and grace are both correct.
(circ) gary and grace are both incorrect.
(circ) gary

Explanation:

Step1: Recall the formula for an angle formed by a secant and a tangent outside a circle.

The measure of an angle formed by a secant and a tangent outside a circle is half the difference of the measures of the intercepted arcs. The formula is \( m\angle = \frac{1}{2}(\text{measure of the larger arc} - \text{measure of the smaller arc}) \). In this case, \( \angle KLM \) is formed by a secant (\( LN \)) and a tangent (\( LJ \)) outside the circle. The intercepted arcs are \( \overset{\frown}{JN} \) (larger arc) and \( \overset{\frown}{KM} \) (smaller arc). So the formula should be \( m\angle KLM=\frac{1}{2}(m\overset{\frown}{JN}-m\overset{\frown}{KM}) \), which is Gary's formula.

Step2: Compare with Grace's formula.

Grace's formula uses the sum of the arcs, which is incorrect for an angle formed outside the circle by a secant and a tangent. The correct formula uses the difference, so Gary is correct.

Answer:

Gary