Question

here is a diagram.
ac is 25 units long.
find the missing values.
type your answers in the boxes.
a. dc=
b. bd=
c. the measure of angle acb= degrees

Explanation:

Step1: Find \(BC\) using Pythagorean theorem

In right - triangle \(ABC\), by Pythagorean theorem \(AB^{2}+BC^{2}=AC^{2}\). Given \(AB = 24\), \(AC = 25\). Then \(BC=\sqrt{AC^{2}-AB^{2}}=\sqrt{25^{2}-24^{2}}=\sqrt{(25 + 24)(25 - 24)}=\sqrt{49}=7\).

Step2: Use similarity of triangles

Triangles \(ABC\) and \(BDC\) are similar. \(\triangle ABC\sim\triangle BDC\) (by AA similarity, \(\angle ABC=\angle BDC = 90^{\circ}\) and \(\angle C=\angle C\)).
We know that \(\sin A=\frac{BC}{AC}=\frac{7}{25}\). Also, in \(\triangle BDC\), \(\sin C=\frac{BD}{BC}\). In \(\triangle ABC\), \(\angle A+\angle C = 90^{\circ}\), \(\sin A=\cos C\).
Another way: Using the formula for the altitude \(BD\) in a right - triangle \(ABC\) with legs \(a = 24\), \(b = 7\) and hypotenuse \(c = 25\). The area of \(\triangle ABC\) is \(S=\frac{1}{2}\times AB\times BC=\frac{1}{2}\times AC\times BD\).
\(\frac{1}{2}\times24\times7=\frac{1}{2}\times25\times BD\), so \(BD=\frac{24\times7}{25}=\frac{168}{25}=6.72\).
For \(DC\), using \(BC^{2}=BD^{2}+DC^{2}\) (from right - triangle \(BDC\)) and also from the property of right - triangle \(ABC\) ( \(BC^{2}=AC\times DC\) (geometric mean theorem: in a right - triangle, the altitude \(BD\) to the hypotenuse \(AC\) gives \(BC^{2}=AC\times DC\))
Since \(BC = 7\) and \(AC = 25\), then \(DC=\frac{BC^{2}}{AC}=\frac{49}{25}=1.96\).
For \(\angle ACB\), \(\sin\angle ACB=\frac{AB}{AC}=\frac{24}{25}\), \(\angle ACB=\sin^{-1}(\frac{24}{25})\approx73.74^{\circ}\) or using \(\cos\angle ACB=\frac{BC}{AC}=\frac{7}{25}\), \(\angle ACB=\cos^{-1}(\frac{7}{25})\approx73.74^{\circ}\)

Answer:

a. \(DC = 1.96\)
b. \(BD = 6.72\)
c. The measure of angle \(ACB\approx73.7^{\circ}\)