Question

if bc = 10, ab = 7, and the perimeter of triangle abc is 27, what is the value of bd?

Explanation:

Step1: Find AC

The perimeter of a triangle is the sum of its three sides. For triangle \(ABC\), perimeter \(= AB + BC + AC\). We know \(AB = 7\), \(BC = 10\), and perimeter \(= 27\). So, \(7 + 10 + AC = 27\). Solving for \(AC\): \(AC = 27 - 7 - 10 = 10\). Wait, no, wait, maybe triangle \(ABC\) is isoceles? Wait, no, looking at the diagram, \(CD\) is a perpendicular from \(C\) to \(AB\), so \(CD\) is the altitude, and \(D\) is on \(AB\). So triangle \(ABC\) has \(AB\) as a side, \(BC\) and \(AC\) as the other two sides. Wait, maybe I made a mistake. Let's re - calculate:

Perimeter \(P=AB + BC+AC\), so \(AC=P - AB - BC\). Substituting the values: \(AC = 27-7 - 10=10\). Wait, so \(AC = 10\) and \(BC = 10\), so triangle \(ABC\) is isoceles with \(AC = BC\). Then, the altitude from \(C\) to \(AB\) (which is \(CD\)) will bisect \(AB\) (because in an isoceles triangle, the altitude from the apex to the base bisects the base). Wait, \(AB\) is the base? Wait, \(AB = 7\), so if \(D\) is the foot of the perpendicular from \(C\) to \(AB\), then \(AD=DB\) only if \(AC = BC\). Wait, we found \(AC = 10\) and \(BC = 10\), so yes, triangle \(ABC\) is isoceles with \(AC = BC\), so the altitude from \(C\) to \(AB\) bisects \(AB\). Therefore, \(BD=\frac{AB}{2}\).

Step2: Calculate BD

Since \(AB = 7\), then \(BD=\frac{7}{2}=3.5\). Wait, but let's check again. Wait, maybe I mis - identified the sides. Let's look at the diagram: \(B\) and \(A\) are on a vertical line, \(D\) is between \(B\) and \(A\), and \(C\) is connected to \(B\), \(A\) and \(D\), with \(CD\) perpendicular to \(AB\). So \(AB\) is the segment from \(A\) to \(B\), with \(D\) in between. The perimeter of triangle \(ABC\) is \(AB + BC+AC\). We know \(AB = 7\), \(BC = 10\), perimeter \(= 27\), so \(AC=27-(7 + 10)=10\). So \(AC = BC = 10\), so triangle \(ABC\) is isoceles with \(AC = BC\). Then, the perpendicular from \(C\) to \(AB\) (i.e., \(CD\)) will bisect \(AB\) (by the property of isoceles triangles: in an isoceles triangle, the altitude from the vertex ( \(C\) in this case) to the base ( \(AB\) in this case) bisects the base). So \(D\) is the mid - point of \(AB\), so \(BD=\frac{AB}{2}\). Since \(AB = 7\), \(BD=\frac{7}{2}=3.5\) or \(\frac{7}{2}\).

Answer:

\(3.5\) (or \(\frac{7}{2}\))