# Given a triangle with angle coordinates: A(0,0), B(3,4), C(7,1) Prove that triangleABC is isosceles.

Question
Given a triangle with angle coordinates:
A(0,0), B(3,4), C(7,1)
Prove that $$\displaystyle\triangle{A}{B}{C}$$ is isosceles.

2021-02-11
An isosceles triangle has at least two sides of equal length. Let's determine the length of each side using the distance formula:
$$\displaystyle{d}=\sqrt{{{\left({\underset{{{2}}}{{{x}}}}-{\underset{{{1}}}{{{x}}}}\right)}^{{2}}+{\left({\underset{{{2}}}{{{y}}}}-{\underset{{{1}}}{{{y}}}}\right)}^{{2}}}}$$
Substitute for AB:
$$\displaystyle{A}{B}=\sqrt{{{\left({3}-{0}\right)}^{{2}}+{\left({4}-{0}\right)}^{{2}}}}=\sqrt{{{9}+{16}}}=\sqrt{{25}}={5}$$
Substitute for BC:
$$\displaystyle{B}{C}=\sqrt{{{\left({7}-{3}\right)}^{{2}}+{\left({1}-{4}\right)}^{{2}}}}=\sqrt{{{16}+{9}}}=\sqrt{{25}}={5}$$
Since $$\displaystyle{A}{B}={B}{C},\triangle{A}{B}{C}$$ is isosceles.

### Relevant Questions

A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors $$\displaystyle\hat{{{i}}}$$ and $$\displaystyle\hat{{{j}}}$$.
1. (Enter in box 1) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+{\left({E}{n}{t}{e}{r}\in{b}\otimes{2}\right)}{P}{S}{K}\frac{{m}}{{s}^{{2}}}\hat{{{j}}}$$
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+$$
5. ( Enter in box 5) $$\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{j}}}$$

1. S1 and S2, shown above, are thin parallel slits in an opaqueplate. A plane wave of wavelength λ is incident from the leftmoving in a direction perpendicular to the plate. On a screenfar from the slits there are maximums and minimums in intensity atvarious angles measured from the center line. As the angle isincreased from zero, the first minimum occurs at 3 degrees. Thenext minimum occurs at an angle of-
A. 4.5 degrees
B. 6 degrees
C. 7.5 degrees
D. 9 degrees
E. 12 degrees
To prove: The similarity of $$\displaystyle\triangle{B}{C}{D}$$ with respect to $$\displaystyle\triangle{F}{E}{D}$$.
Given information: Here, we have given that $$\displaystyle\overline{{{A}{C}}}\stackrel{\sim}{=}\overline{{{A}{E}}}\ {\quad\text{and}\quad}\ \angle{C}{B}{D}\stackrel{\sim}{=}\angle{E}{F}{D}$$
A triangle is given with one side equal to 18m (a) and angles laying on this side equal to $$\displaystyle{48}^{\circ}$$ (A) and $$\displaystyle{37}^{\circ}$$ (B). Find the ramaining angle (C) and sides (b and c).
Mg reacts with H+ (aq) according to
$$\displaystyle{M}{g{{\left({s}\right)}}}+{2}{H}+{\left({a}{q}\right)}\to{M}{g}_{{2}}+{\left({a}{q}\right)}+{H}_{{2}}{\left({g}\right)}$$
Suppose that 0.524 g of Mg is reacted with 60.o ml of 1.0 M H+(aq). Assume that the density of the H+ (aq) solution os 1.00 g/ml,and that its specific heat capacity equals that of water. Theinitial and final temperatures are 22.0 degree celsius and 65.8degree celsius.
a) Is the reaction endothermic or exothermic?
b) Calculate $$\displaystyle\triangle{H}$$ of of th reaction. Use correct sigs andgive units.
c) Calculate the $$\displaystyle\triangle{H}$$ of the reaction per mole of magnesium.
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
Solid NaBr is slowly added to a solution that is 0.010 M inCu+ and 0.010 M in Ag+. (a) Which compoundwill begin to precipitate first? (b) Calculate [Ag+] when CuBr justbegins to precipitate. (c) What percent of Ag+ remains in solutionat this point?
a) AgBr: $$\displaystyle{\left({0.010}+{s}\right)}{s}={4.2}\cdot{10}^{{-{8}}}$$ $$\displaystyle{s}={4.2}\cdot{10}^{{-{9}}}{M}{B}{r}$$ needed form PPT
CuBr: $$\displaystyle{\left({0.010}+{s}\right)}{s}={7.7}\cdot{\left({0.010}+{s}\right)}{s}={7.7}\cdot{10}^{{-{13}}}$$ Ag+=$$\displaystyle{1.8}\cdot{10}^{{-{7}}}$$
b) $$\displaystyle{4.2}\cdot{10}^{{-{6}}}{\left[{A}{g}+\right]}={7.7}\cdot{10}^{{-{13}}}$$ [Ag+]$$\displaystyle={1.8}\cdot{10}^{{-{7}}}$$
c) $$\displaystyle{\frac{{{1.8}\cdot{10}^{{-{7}}}}}{{{0.010}{M}}}}\cdot{100}\%={0.18}\%$$
To prove: The similarity of $$\displaystyle\triangle{N}{W}{O}$$ with respect to $$\displaystyle\triangle{S}{W}{T}$$.