Question

find each length below.
(a) in the figure below, two chords intersect inside the circle at point h.
suppose that hn = 35, hj = 14, and hm = 7. find hk.
(b) in the figure below, a tangent and a secant are drawn to a circle from exterior point v.
suppose that vg = 12 and vc = 7.2. find cd.

Explanation:

Part (a)

Step1: Recall Chord Intersection Theorem

When two chords intersect inside a circle, the products of the segments of each chord are equal. So, \( HM \times HK = HJ \times HN \).

Step2: Substitute Known Values

We know \( HM = 7 \), \( HJ = 14 \), \( HN = 35 \). Let \( HK = x \). Then the equation becomes \( 7 \times x = 14 \times 35 \).

Step3: Solve for \( x \)

First, calculate the right - hand side: \( 14\times35 = 490 \). Then, solve for \( x \) from \( 7x = 490 \). Divide both sides by 7: \( x=\frac{490}{7}=70 \).

Step1: Recall Tangent - Secant Theorem

If a tangent segment \( VG \) and a secant segment \( VD \) (where \( VD=VC + CD \)) are drawn from an external point \( V \) to a circle, then \( VG^{2}=VC\times VD \). Let \( CD = y \), so \( VD=VC + y=7.2 + y \).

Step2: Substitute Known Values

We know \( VG = 12 \) and \( VC = 7.2 \). Substitute into the formula: \( 12^{2}=7.2\times(7.2 + y) \).

Step3: Solve the Equation

First, calculate \( 12^{2}=144 \). So the equation is \( 144 = 7.2\times(7.2 + y) \). Divide both sides by 7.2: \( \frac{144}{7.2}=7.2 + y \). \( 20=7.2 + y \). Then, subtract 7.2 from both sides: \( y = 20 - 7.2=12.8 \).

Answer:

\( HK = 70 \)

Part (b)